•  THEORY    OF   PHYSICS' 


UNIVERSITY  OF  CALIFORNIA 

DEPARTMENT  OF  PHYSICS 


BY 


JOSEPH  S.  AMES,  Pn.D. 

M 

ASSOCIATE     PROFESSOR     OF     PHYSICS     AND 

SUB-DIRECTOR    OF    THE    PHYSICAL    LABORATORY   IN 

JOHNS    HOPKINS   UNIVERSITY 


NEW    YORK   AND    LONDON 

HARPER    &    BROTHERS    PUBLISHERS 
1897 


022: 


Copyright,  1897,  by  HARPER  &  BROTHERS. 

All  rig/its  reserved. 


PREFACE 


To  present  successfully  the  subject  of  Physics  to  a 
class  of  students,  three  things  seem  to  me  necessary:  a 
text-book;  a  course  of  experimental  demonstrations  and 
lectures,  accompanied  by  recitations ;  and  a  series  of 
laboratory  experiments,  .mainly  quantitative,  to  be  per- 
formed by  the  students  themselves  under  the  direction 
of  instructors.  I  place  "text-book"  first,  because  for 
many  reasons  I  believe  it  to  be  the  most  important  of 
the  three.  None  but  advanced  students  can  be  trusted 
to  take  accurate  and  sufficient  notes  of  lectures ;  and  a 
text-book  which  states  the  theory  of  the  subject  in  a 
clear  and  logical  manner  so  that  recitations  can  be  held 
on  it,  seems  to  me  absolutely  essential.  Another  great 
advantage  of  a  text-book  is  that  a  student  provided 
with  one  does  not  need  to  take  notes  on  the  lectures, 
and  so  can  give  his  undivided  attention  to  the  expla- 
nations and  demonstrations  which  are  being  presented. 

It  has  been  my  aim  in  writing  this  book  to  give  a 
concise  statement  of  the  experimental  facts  on  which 
the  science  of  Physics  is  based,  and  to  present  with 
these  statements  the  accepted  theories  which  correlate 
or  "explain"  them.  I  have  in  no  case  resorted  know- 
ingly to  an  antiquated  or  incorrect  theory  for  the  sake 

673269 


iv  PREFACE 

of  ease  of  demonstration ;  and  I  have  constantly  tried  to 
show  the  connections  between  different  phenomena,  and 
to  reduce  the  general  laws  of  nature  to  as  small  a  num- 
ber as  possible.  Comparatively  little  attention  is  given 
in  the  text  to  the  description  or  discussion  of  the  great 
fundamental  experiments  of  Physics;  because  this  can 
be  given  in  the  class  demonstrations ;  and,  in  any  case, 
this  side  of  the  subject  deserves  consideration  in  a  sep- 
arate book.  The  central  thought  of  the  present  book  is 
the  theory  of  the  experiments,  and  their  "explanation" 
in  terms  of  more  fundamental  ideas  and  principles.  It 
is  for  this  reason  that  I  have  chosen  the  title  "Theory 
of  Physics."  I  am  not  at  all  .certain  that  the  choice 
is  a  happy  one ;  but  it  at  least  has  the  merit  of  empha- 
sizing what  I  have  desired  to  make  the  chief  character- 
istic of  the  book. 

As  any  teacher  knows,  it  is  well-nigh  impossible  to 
give  courses  of  lectures  on  any  subject  without  from 
time  to  time  adopting  modes  of  expression  and  expla-. 
nation,  even  outlines  of  certain  theories,  from  various 
books  and  articles ;  and  I  am  perfectly  conscious,  in  my 
own  case,  of  having  received  great  assistance  in  number- 
less ways  from  many  writers.  The  present  book  is  in 
the  main  simply  a  revision  of  a  course  of  lectures  which 
I  have  been  giving  for  some  years;  and  I  am  certain 
that  many  ideas  which  are  not  my  own  have  crept  in. 
They  are  now,  however,  so  assimilated,  even  in  my  mind, 
that  it  is  absolutely  impossible  for  me  to  distinguish 
that  which  is  strictly  original  from  that  which  is  bor- 
rowed. If  any  one  wishes  to  claim  any  portion  of  the 
book  as  his  and  not  mine,  I  shall  be  the  first  to  try  to 


PREFACE  V 

recognize  his  claim ;  but  I  can  give  the  strictest  assur- 
ance that  in  no  case  have  I  deliberately  used  even  a 
sentence  or  a  single  idea  which  belongs  to  another. 

The  present  text-book  is  designed  for  those  students 
who  have  had  no  previous  training  in  Physics,  or  at 
the  most  only  an  elementary  course;  and  it  should,  then, 
be  adapted  to  junior  classes  in  colleges  or  technical 
schools.  The  entire  subject  as  here  presented  may  be 
easily  studied  in  a  course  lasting  for  the  academic  year 
of  nine  months.  Although  there  is  a  close  connection 
between  the  various  sections,  I  have  tried  to  divide  the 
subject  in  such  a  way  that  it  need  not  all  be  considered 
in  one  continuous  course ;  for  in  many  colleges  only 
selected  portions,  e.  g.  Heat  arid  Light,  are  taken  up  at 
a  time.  I  ought,  perhaps,  to  state  the  reasons  which 
have  led  me  to  make  the  divisions  and  to  arrange  them 
as  I  have ;  but  in  reality  I  have  but  one  reason  :  I  have 
constantly  had  in  mind  the  needs  of  the  student,  and 
have  tried  in  every  way  to  treat  the  subject  so  as  to 
make  it  clearest  to  him. 

I  wish  to  express  my  sense  of  obligation  to  my  as- 
sistant, Mr.  N.  E.  Dorsey,  who  has  kindly  read  my  entire 
manuscript  and  called  my  attention  to  many  points  de- 
manding correction ;  and  also  to  Mr.  C.  L.  Eeeder,  who 
has  made  all  the  drawings  most  skilfully  and  promptly. 

J.  S.  AMES. 
JOHNS  HOPKINS  UNIVERSITY, 

BALTIMORE,  October,  1896. 


CONTENTS 


INTRODUCTION 

PAGE 

Physics l 

Matter  and  its  Properties 

Conservation  of  Matter 

Motion 

Fundamental  Units 

Mechanics 


BOOK   I 
MECHANICS   AND   PROPERTIES   OF   MATTER 

CHAPTER  I 

KINEMATICS 

10.  Motion  in  General 13 

11-15.  Translation H-21 

16.  Rotation 21,22 

17.  Numerical  Value  of  an  Angle 22 

18.  Translation.     1.  Motion  in  a  Straight  Line 23,24 

19.  2.  Motion  in  a  Circle 25, 26 

20.  Combination  of  1  and  2 27 

21.  3.  Harmonic  Motion 27,28 

22.  Rotation.     1.  Motion   around  an  Axis  whose  Direction 

does  not  change      28 

23.  2.  Motion  of  a  Spinning-Top 29 

24.  Combination  of  1  and  2 

25.  3.  Harmonic  Motion 30 


viii  CONTENTS 


CHAPTER  II 

DYNAMICS 

ARTICLE  PAGE 

26.  Measurement  of  Mass  by  means  of  Inertia  .....  31,  32 

27.  Matter  in  Translation.    Conservation  of  Linear  Momentum  33 

28.  Special  Cases.     1.  One  Body 34 

29.  2.  Two  Bodies 34, 35 

30-32.      Force.     Newton's  "  Laws  of  Motion  " 35-38 

33.  Special  Cases.     1.  Illustration  of  Action  of  Earth    ...  38 

34.  2.  Atwood's  Machine 39 

35.  3.   Whirling  Table 40 

36.  4.  Parallelogram  of  Forces 41, 42 

37.  5.  Inclined  Plane 42 

38.  39.      Centre  of  Inertia 42-47 

40.  Special  Cases 47,  48 

41.  Matter   in   Rotation.     Conservation  of  Moment   of   Mo- 

mentum        49, 50 

42.  43.  Special  Cases.     1.  Single  Particle.     Moments      ....  50-52 

44.  2.  Fixed  Axis 53 

45.  Illustrations,     a.  Disc 54 

46.  b.  Two  Discs 54 

47.  c.  Inclined  Plane     .........  55 

48.  d.  Top 56 

49, 50.      Centre  of  Inertia        56-58 

51.  Harmonic  Motion 58-60 

52.  Motion  in  General  of  a  Rigid  Body     .*...„...  60 

53.  a.  Non-Parallel  Coplanar  Forces 61-63 

54-56.                   b.  Parallel  Forces .....  64-66 

57, 58.                    c.  Couples 67 

59.  Equilibrium 68 

60.  Special  Cases.     1 .  Triangle  of  Forces 68, 69 

61-64.                                    2, 3.  Rigid  Body 69-72 

65.  Stability  of  Equilibrium 73 

66.  Principle  of  Stable  Equilibrium 74 

67.  Work  and  Energy 75,  76 

68.  Measure  of  Work  and  Energy    ....  ....  76*-80 

69.  Transfer  of  Energy 80-82 

70.  Machines 83 

71.  Lever.     Chemical  Balance 83-85 

72.  Pulley 85-88 

73.  Screw 88, 89 

74.  Wave-Motion 90-94 

75.  Wave-Fronts.     Secondary  Waves  .........  94, 95 


CONTENTS  ix 


CHAPTER  III 

GRAVITATION 

ARTICLE  PAGE 

76.  Weight 96, 97 

77.  Universal  Gravitation 97-99 

78.  Centre  of  Gravity 99-101 


CHAPTER  IV 

PROPERTIES    OF    SIZE    AND    SHAPE    OF    MATTER 

79.  Introduction.     Forms  of  Matter.     Strains 102-104 

80.  Properties  of  Solids 104 

81.  Change  in  Volume .  104 

82.  Change  in  Shape 105,  106 

83.  Young's  Modulus 107-109 

84.  Properties  of  Liquids 109 

85-88.             Liquids  at  Rest 109-112 

89,90.                          Pressure  due  to  Weight 112-115 

91.  Measurement  of  Density  of  a  Liquid   .     .     .  115-117 

92.  Archimedes'  Principle 117 

93.  Measurement  of  Density  of  a  Solid      ...  118 

94.  Floating  Bodies 119 

95.  Work  required  to  Change  the  Volume  of  a 

Liquid 120 

S6.                    Liquids  in  Motion 121,122 

97.  Capillarity 123-128 

98.  Properties  of  Gases 128 

99.  Gases  at  Rest 128,129 

100-102.                     Atmospheric  Pressure.     Pumps.     Siphon      .  129-132 

103.  Measurement  of  Pressure 132,133 

1 04.  Work  required  to  Change  the  Volume  of  a  Gas  1 34 

105.  Gases  in  Motion 134 

106.  Gases  as  Distinct  from  Liquids 134 

107.  Dalton's  Law 135 

108.  Boyle's  Law 135 

109.  Elasticity  of  a  Gas 136 

110,111.                     Air-Pumps 137-140 


CONTENTS 


BOOK   II 
SOUND 


ARTICLE 


INTRODUCTION 
112.         Description  of  Phenomena      .    .    .    . 


PAGE 


145 


113. 

114. 
115. 


CHAPTER  I 

VIBRATIONS 

Detection  and  Nature  of  the  Vibrations 
Periodic  Vibrations.     Characteristics  . 
Measurement  of  Period  of  Vibration    . 


146,  147 
147, 148 
149, 150 


CHAPTER  II 

SOUND   WAVES 

116-118.     Nature  of  the  Wares 151-153 

119.              Velocity 153,154 

120,121.     Composition  of  Waves.     Characteristics 154-156 

CHAPTER  III 

SOUND   SENSATION 

122,123.      Perception  of  Sound.     The  Ear 157,158 

124,125.      Musical  Notes 158,159 

126, 127.      Pitch.     Doppler's  Principle 159-161 

128.             Intensity 161 

129,130.      Quality 161 

131-133.      Combinations.     Beats  163-165 


CHAPTER  IV 

REFLECTION   AND    REFRACTION 


134.  General  Description.     "Refraction 

135.  Reflection.    Transverse  Waves    . 
136-138.  1 .  From  a  Fixed  End    .     . 

139.  2.  From  a  Free  End      .     . 

140.  Longitudinal  Waves 


166 

167,168 

169-171 

171 

172 


CONTENTS  xi 


CHAPTER  V 

VIBRATING    BODIES 

ABTICLE  PAGE 

141.  Stationary  Vibrations          173 

142.  Transverse  Vibrations  of  a  Cord 173-175 

143.  Longitudinal  Vibrations  in  a  Cord  or  Wire       ....  176 

144.  Longitudinal  Vibrations  of  Rods 177 

145.  Transverse  Vibrations  of  Rods         177 

146.  Vibrations  of  Columns  of  a  Gas        178 

147.  1.  Closed  Pipes         178,179 

148.  2.  Open  Pipes 179-181 

149.  Vibrations  of  Plates  and  Membranes 181 

150.  Vibrations  of  Bells          181 

151.  The  Human  Voice 182 

152.  Connection  between  Vibrating  Body  and  Wave  Emitted  182 


CHAPTER  VI 

VELOCITY   OF   SOUND 

153.  Direct  Method .*....  183 

154.  Indirect  Method 183 

155.  Velocity  in  a  Column  of  Gas 184 

156.  Velocity  in  a  Column  of  Liquid       185 

157.  Velocity  in  a  Solid  Wire  or  Rod 185 

158.  Kundt's  Method  186 


CHAPTER  VH 

HARMONY   AND   MUSIC 

159.  Musical  Sounds 188 

160.  Numerical  Relations 188 

161.  Harmony  and  Discord 183-190 

762.         Musicaf  Scales  .                                                         .     .     .  190,191 


BOOK   III 

HEAT 

INTRODUCTION 

163.  Nature  of  Heat-Effects .       195,196 

164.  Sources  of  Heat-Energy 196 


xii  CONTENTS 


CHAPTER  I 

HEAT-EFFECTS  —  TEMPERATURE 

ARTICLE  PAGE 

165.  Change  in  Volume 197 

166.  Change  in  Temperature 198 

167.  Molecular  Changes 199 

168.  Temperature.     Thermometers 200-203 


CHAPTER  II 

CHANGES    IN    VOLUME 

169.  Solids.     Linear  Expansion 204 

170.  Application  of  Principle  of  Stable  Equilibrium    .  206 

171.  Cubical  Expansion 206 

172.  Liquids.     Cubical  Expansion 207 

173.  Water 208 

174.  Application  of  Principle  of  Stable  Equilibrium    .  208 

175.  Barometric  Correction 208,209 

176.  Measurement  of  Coefficient  of  Expansion   .     .     .  210,211 

177.  Gases.     Cubical  Expansion 212 

178.  Changes  of  Pressure  and  Temperature  ....  212,215 

179.  Measurement  of  Coefficient  of  Expansion   .     .     .  215 


CHAPTER   III 

CHANGE    IN    TEMPERATURE 

180.  Thermal  Unit 216 

181.  Specific  Heat 217 

182.  Specific  Heat  of  Gases 218,219 

183.  Dulong  and  Petit's  Law 220 

184.  Measurement  of  Specific  Heat 220-224 


CHAPTER  IV 

MOLECULAR   CHANGES 

185.  Expansion 225 

186.  Internal  Work  in  a  Gas 225,226 

187.  Fusion.     Fusion-Point .     '.    •  226, 227 

188.  Change  in  Volume 228,229 

189.  Latent  Heat  of  Fusion     ..........  229 

190.  Effect  of  Dissolved  Substances 230 


CONTENTS  xiii 

ARTICLE  PAGE 

191.  Evaporation.     Vapors , 231 

192.  Laws  of  Saturated  Vapor 231,232 

193.  The  Statical  Method 233 

194.  The  Dynamical  Method 234-236 

195.  Latent  Heat  of  Evaporation 236,237 

196.  Dalton's  Law  for  Mixtures  ...  ...  238 

197.  Atmospheric  Moisture 238 

198.  Spheroidal  State 238 

199.  .              Effect  of  Dissolved  Substances 238 

200.  Liquefaction  of  Gases.     Isothermals 239-242 

201.  Sublimation 242 

202.  Dissociation  and  Combination 242, 243 

203.  Solution    .  243,244 


CHAPTER  V 

TRANSFER    OF    HEAT-ENERGY 

• 

204.  Convection 245 

205.  Conduction 246 

206, 207.  Kadiation.     Absorption.     Emission 247-251 

208.  Flow  of  Heat-Energy 251 


CHAPTER  VI 

KINETIC    THEORY    OF    MATTER 

209, 210.      General  Statement 253-255 

211.  Kinetic  Theory  of  Gases 255-261 


CHAPTER   VII 

THERMODYNAMICS 

First  Principle    .     . 262 

Second  Principle 262-264 


BOOK    IV 
ELECTRICITY  AND   MAGNETISM 

INTRODUCTION 
214.         Properties  of  the  Ether .    .    .  267 


XIV 


CONTENTS 


CHAPTER  I 


GENERAL    PROPERTIES    OF    ELECTRIC    CHARGES 

AETICLE  PAGE 

215.  Production  of  Charges 269 

216.  Positive  and  Negative  Charges 270,  271 

217.  "  Specific  Attraction "  of  Matter  for  Electricity     .     .  271 

218.  Quantities  of  Electricity 272 

219.  Law  of  Electrostatic  Force 273 

220.  Conductors  and  Dielectrics 274-277 

221.  Energy  of  Electrostatic  Field 277-279 

222.  Lines  of  Electrostatic  Force 279-282 

CHAPTER  II 

ELECTRIC    POTENTIAL    AND    INDUCTION 

223.  Electric  Potential  .    * 283, 284 

224.  Electric  Forces 284 

225.  Potential  at  a  Point 285 

226.  Potential  of  a  Conductor 286 

227.  Distribution  of  Charges  on  Conductors 286 

228.  Equipotential  Surfaces 287 

229, 230.  Electrostatic  Induction 288-291 

231.  Charging  by  Induction 291 

232.  Faraday's  Ice-Pail  Experiment 291 

233.  Faraday's  Tubes 293 

234.  Shielding  by  Closed  Conductors 294 

235.  Condensers 294-297 

236.  Discharge  of  Condensers 297, 298 

237.  Capacity 299,300 

238.  Energy  of  Electrification 301 

239.  Electrostatic  Measurements 302 

CHAPTER  III 

ELECTRIC    CONDUCTION 

240.  Electric  Current 304-307 

241.  Thermo-Electric  Currents      ........  307-310 

242.  Primary  Cells.     Energy 310-315 

243.  Nature  of  Conductors 315 

244-246.  Electrolysis 316-322 

247-249.  Dissociation  Theory  of  Electrolytes 322-326 

250.  Discharge  through  Gases 326-328 


CONTENTS  XV 


CHAPTER  IV 

PROPERTIES    OF    STEADY    ELECTRIC    CURRENTS 

ARTICLE  PAGE 

251.  Uniformity  of  Current 329 

252.  Heating  Effect 329, 330 

253.  Magnetic  Effect 331,332 

254.  Ohm's  Law 333, 334 

255.  Applications.     1.  Conductors  in  Series 335 

256.  2.  Conductors  in  Parallel       ....  335, 336 

257.  3.  Wheatstone's  Bridge 337, 338 

258.  4.  Heating  Effect 339 


CHAPTER  V 

GENERAL    PROPERTIES   OF    MAGNETS    AND    MAGNETIC    FIELDS 

259, 260.      Definitions.     General  Properties 340-342 

261,262.      Polarity.     Equality  of  Poles 342,343 

263.  Molecular  Nature  of  Magnetism 343-345 

264.  Unit  Poles.     Law  of  Force 345, 346 

265.  Attraction  and  Repulsion ...  346 

266.  Magnetic  Lines  of  Force ...  347 

267.  Magnetic  Induction 348-350 

268.  Magnetic  Moment        351 

269-271.     Measurement  of  Magnetic  Quantities 352-355 

CHAPTER  VI 

MAGNETISM    OF    THE    EARTH 

272.  Magnetic  Elements 356 

CHAPTER  VII 

MAGNETIC    PROPERTIES    OF    STEADY    ELECTRIC    CURRENTS 

273.  Magnetic  Field  around  a  Current 359 

274.  Electro-Magnetic  Force 361 

275.  Ampere's  Theory  of  Magnetism 362 

276.  Electric  Motors 363 

277.  Unit  Current 363 

278.  Tangent  Galvanometer 365 

279.  280.      Electro-Magnetic  System  of  Units 367 

281.             Energy  of  a  Magnetic  Field 369 


XVI 


CONTENTS 


CHAPTER  VIII 


INDUCED    CURRENTS 
ARTICLE 

282.  General  Properties 

283-288.      Special  Cases 

289. 

290. 

291. 

292. 

293. 


Transformer  or  Induction  Coil     . 

Dynamo  and  Motor 

Telephone 

Microphone 

Weber's  Theory  of  Diamagnetism 


PAGE 
371 

372-377 
378 
381 
383 
384 
385 


CHAPTER  IX 

GENERAL   PROPERTIES    OF   A   MAGNETIC    FIELD 


294.  Evidence  of  Rotation       .     .    .     . 

295.  Rotation  of  Plane  of  Polarization 

296.  The  Hall  Effect 


386 
386 
386 


BOOK    V 
LIGHT 


INTRODUCTION 


297.  Description  of  Phenomena 

298,299.     Ether- Waves       .... 


391 
392 


CHAPTER  I 


THE    WAVE    THEORY 


300.  Young's  Interference  Experiment 

301.  Velocity  of  Light 

302.  Rectilinear  Propagation  .... 

303.  Shadows 

304.  Pin-Hole  Images 


395 
398 
401 
403 
405 


CONTENTS 


xvn 


CHAPTER  II 


REFLECTION 
ARTICLE 

305.  Mirrors 

306.  Plane  Waves  Incident  ou  a  Plane  Mirror       .     .     . 

307.  Rotating  Plane  Mirror 

308.  Spherical  Waves  Incident  on  a  Plane  Mirror     .     . 

309.  Inclined  Mirrors 

310-316.  Spherical  Waves  Incident  on  a  Spherical  Mirror    . 

317.  Spherical  Aberration       


PAGE 
407 
407 
409 
410 
412 

413-420 
420 


CHAPTER  III 

REFRACTION 

318.  Introduction 422 

319.  Plane  Waves  Refracted  at  a  Plane  Surface  ....  422 

320.  Total  Reflection 427 

321.  Plate  with  Plane  Parallel  Faces 428 

322.  Prism 429 

323-325.      Spherical  Waves  Refracted  at  a  Plane  Surface  ...  431 

326.  Plane  Waves  Refracted  at  Spherical  Surfaces   .     .     .  434 

327.  Spherical  Waves  Refracted  at  a  Spherical  Surface     .  436 

328-339.      Lenses 440-450 

340-343.      Combinations  of  Lenses :  Microscopes,  Telescopes,  etc.  450-454 


CHAPTER  IV 


DISPERSION  —  SPECTRA 


344.  Dispersion 

345.  Pure  Spectrum        

346.  Anomalous  Dispersion     .... 
347  Achromatism 

348.  Direct- Vision  Spectroscope      .     . 

349.  Ordinary  Spectroscope     .... 
350,351.  Various  Kinds  of  Spectra    .     .     . 

352.  Frailnhofer's  Lines 

353-355.  Absorption :  Surface  and  Ordinary 

356,  357.  Fluorescence,  Phosphorescence 

358.  Reflection  by  Fine  Particles     .     . 


455 
456 
457 
458 
460 
461 

462-464 
464 
465 
466 
466 


XV111 


CONTENTS 


CHAPTER   V 


COLORS  —  COLOR-SENSATION 

ARTICLE  PAGE 

359.  Complementary  Colors 468 

360.  Colors  of  Objects 469 

361.  Perception  of  Color 470 

CHAPTER  VI 

INTERFERENCE  —  DIFFRACTION 

362.  Interference  Due  to  Two  Sources 471 

363.  Colors  of  Thin  Plates      . 475 

364.  Diffraction  by  an  Edge 477 

365.  Diffraction-Grating 479 

CHAPTER  VII 

DOUBLE    REFRACTION 

366-369.     Description  of  Phenomenon 483, 484 

370.  Nicol's  Prism 485 

371.  Property  of  Tourmaline       485 

CHAPTER  VIII 

POLARIZATION 

372.  Polarization  by  Reflection 487 

373.  Polarization  by  Double  Refraction 489 

374.  Circularly  Polarized  Waves 492 

375.  Interference  of  Polarized  Waves ._  493 

376.  Colors  due  to  Polarization 493 

377.  Rotation  of  the  Plane  of  Polarization 495 

INDEX   .  ....        497-513 


THEORY    OF    PHYSICS 


INTRODUCTION 

1.  Physics.  As  we  observe  the  world  of  nature  around 
us,  we  are  constantly  receiving  sensations  caused  by  phe- 
nomena which  are  independent  of  ourselves.  Often  we 
are  able  to  determine  the  exact  conditions  which  produce 
a  certain  phenomenon ;  and,  as  the  result  of  observations 
which  have  been  carried  on  for  many  hundred  years,  it 
may .  be  stated  that  there  is  every  evidence  for  believing 
that,  if  the  exact  conditions  are  ever  reproduced,  the  same 
result  always  follows.  This  is  equivalent  to  saying  that 
there  are  certain  laws  in  nature  which  are  entirely  inde- 
pendent of  time,  of  place,  and  of  men.  It  is  the  aim  of 
Natural  Science  to  discover  these  laws.  Physics,  a  branch 
of  natural  science,  is  concerned  with  only  a  limited 
number  of  these  phenomena  of  nature  and  the  laws 
according  to  which  they  take  place ;  but  it  is  practically 
impossible  to  give  a  definite  limitation  to  its  field.  Physics 
includes  the  study  of  many  properties  of  matter,  such  as 
its  inertia,  its  weight,  and  its  form ;  also  the  consideration 
of  the  phenomena  of  sound,  of  heat,  of  light,  of  electricity 
and  magnetism.  Physics,  of  course,  comes  closely  in  con- 
tact with  many  other  natural  sciences,  especially  with 
Chemistry  and  Astronomy. 


2  THEORY  OF  PHYSICS  [INT. 

2.  Matter.  The  name  "  matter "  is  ordinarily  given  to 
all  those  substances  which  in  any  way  appeal  to  the 
senses.  This  must  not  be  regarded  as  a  definition ;  for 
that  can  be  given  only  in  a  description  of  the  properties  of 
matter.  The  word  "matter"  conveys  a  fairly  definite 
idea  to  our  minds  simply  as  the  consequence  of  our  daily 
experiences  ;  but  if  we  analyze  our  ideas  of  it,  we  see 
that  among  them  all  there  are  three  which  are  apparently 
independent  of  each  other,  and  which  in  this  sense  are 
fundamental  Our  original  consciousness,  though,  of  these 
,  fur  da  mental ,  properties  of  matter  depends  upon  only  one 
•of  oar  senses,  that  of  muscular  resistance. 

If  we  in  any  way  alter  the  motion  of  a  piece  of  matter, 
we  are  conscious  of  a  certain  muscular  exertion.  Of 
course  this  same  change  in  the  motion  of  the  matter  could 
be  produced  by  other  causes  than  our  muscles  ;  but  our 
own  conception  of  matter  depends  on  our  muscular  sense. 
Illustrations  of  this  property  are  afforded  when  we  throw 
or  stop  a  ball,  when  we  open  a  door  or  push  a  barrel  along 
a  floor  ;  and  it  has  received  the  name  "  inertia." 

If  we  raise  a  piece  of  matter  from  the  floor  to  the  top  of 
a  table,  —  that  is,  if  we  move  a  portion  of  matter  further 
from  the  earth,  —  we  are  conscious  of  muscular  exertion. 
And,  so  far  as  we  can  see,  there  is  no  connection  between 
this  fact  and  the  property  of  inertia.  We  shall  learn  later 
that  this  consciousness  of  muscular  effort  when  a  piece  of 
matter  is  raised  from  the  earth  is  only  a  special  case  of  a 
much  more  general  property  ;  for  there  seems  to  be  reason 
for  believing  that  if  we  were  to  increase  the  distance  apart 
of  any  two  portions  of  matter,  however  small  or  large,  we 
should  be  conscious  of  muscular  sensation,  if  only  our 
senses  could  be  delicate  enough.  This  property  of  matter 
is  called  its  "  weight." 

If  we  alter  in  any  way  the  shape  or  volume  of  a  portion 
of  matter,  we  are/  in  general,  conscious  of  muscular  exer- 
tion. Illustrations  of  this  fact  are  given  when  we  stretch 


3]  INTRODUCTION  3 

or  twist  a  wire,  squeeze  out  thin  a  piece  of  putty,  compress 
a  gas  in  a  rubber  bag,  etc.  Various  names  have  been  given 
to  this  property,  depending  upon  the  particular  effect  pro- 
duced. These  will  be  discussed  later  under  the  names 
"elasticity,"  "plasticity,"  "ductility,"  etc. 

Any  substance  having  these  three  properties  is  called 
matter.  There  is  no  reason  for  thinking  that  they  are  all 
independent  of  each  other ;  but  at  present  we  have  no 
evidence  which  will  allow  us  to  regard  one  property  as  a 
consequence  of  the  others,  except  in  the  case  of  gases, 
where  we  can  show  that  the  elasticity  is  a  direct  result  of 
inertia. 

It  must  be  carefully  borne  in  mind  that,  although  the 
ideas  of  the  properties  of  matter  come  entirely  through  the 
ordinary  senses,  we  may  have  conclusive  evidence  as  to 
the  nature  of  other  things  which  in  no  way  appeal  to 
the  senses.  We  shall  have  several  illustrations  of  this 
in  the  course  of  our  study. 

3.  Forms  of  Matter.  That  different  kinds  of  matter 
appear  differently  to  our  senses  is  evident  to  all.  Perhaps 
the  most  obvious  fact  is  that  they  may  have  different 
volumes  and  shapes ;  and  it  is  by  means  of  these  properties 
that  we  ordinarily  distinguish  between  them.  The  names 
"solid,"  "liquid,"  and  "gas"  have  been  given  to  certain 
forms  of  matter,  whose  properties  differ  widely. 

We  think  of  a  "solid"  as  having  a  definite  size  and 
shape  which  are  independent  of  its  position  or  condition, 
in  general.  Thus  a  stone  may  rest  on  a  table  or  in  one's 
hand,  or  it  may  be  falling  through  the  air;  it  will  have  the 
same  volume  and  the  same  shape. 

A  "  liquid,"  however,  although  it  has  a  definite  volume, 
assumes  the  shape  of  the  vessel  which  contains  it.  If 
water  is  poured  out  of  one  pitcher  into  another  of  different 
shape,  it  will  occupy  the  same  volume  in  the  two  vessels, 
but  the  shape  will  be  entirely  changed.  Unless  the  liquid 
completely  fills  the  containing  vessel,  there  will  always  be 


4  THEORY  OF  PHYSICS  [INT. 

a  surface  of  separation  between  the  liquid  and  the  air  or 
whatever  gas  fills  the  rest  of  the  vessel.  This  surface  is 
called  the  "  free  surface,"  and  has  some  most  interesting 
properties,  one  of  which  is  that  it  enables  liquids  to  form 
"  drops  "  when  they  escape  from  the  containing  vessel  and 
fall  towards  the  earth. 

A  "gas"  if  placed  inside  a  closed  vessel  diffuses  uni- 
formly throughout  the  entire  space ;  so  that  its  volume 
as  well  as  its  shape  depends  solely  on  the  vessel,  not  on 
the  gas  itself.  Most  gases  are  transparent,  and  so  cannot 
be  seen ;  but  their  properties  may  be  studied  in  other  ways. 

As  we  shall  see  later  on,  these  states  of  matter  —  solid, 
liquid,  and  gas  —  are  only  three  particular  ones  out  of  a 
great  many.  There  is  an  indefinite  number  of  intermediate 
states  which  cannot  be  called  solid,  liquid,  or  gaseous. 
Gases  and  liquids  are  sometimes  called  "  fluids,"  because 
they  possess  in  common  the  property  of  'flowing;  that  is, 
they  assume  the  shape  of  the  containing  vessel. 

All  these  forms  of  matter  can  be  proved  to  have  inertia, 
weight,  and  elasticity ;  although  this  fact  is  not  so  evident 
at  first  sight  in  the  case  of  gases.  This  is  simply  owing  to 
the  difficulty  of  detecting  small  effects  by  means  of  our 
senses.  If  large 'amounts  of  a  gas  are  taken,  or  if  instru- 
ments more  delicate  than  our  senses  are  used,  the  ordinary 
properties  of  matter  may  be  easily  shown  to  hold  for  gases 
as  well  as  for  solids  and  liquids. 

4.  Divisibility  of  Matter.  That  there  are  spaces  between 
the  portions  of  matter  making  up  a  substance  is  shown  by 
many  facts  of  daily  experience.  This  is  self-evident  in  the 
case  of  wood,  cork,  sandstone,  etc.,  and  in  all  gases.  Also, 
gases  can  in  many  cases  pass  through  solids,  as  carbon- 
dioxide  through  red-hot  iron.  If  ordinary  salt  is  thrown  in 
water,  it  is  evidently  broken  up  into  parts  which  are  so 
small  that  they  can  pass  between  the  small  portions  of  the 
water ;  for  the  bulk  of  the  water  and  salt  when  mixed  is 
not  the  sum  of  the  volumes  of  the  two  separately. 


4]  INTRODUCTION  5 

We  can  imagine,  then,  a  portion  of  matter  divided  into 
parts,  and  these  in  turn  into  still  finer  parts;  and  so  on, 
each  of  the  parts  having  the  same  properties  as  the  original 
substance.  Thus  a  piece  of  chalk  can  be  broken  into  two 
pieces  of  chalk,  each  of  these  into  two  other  pieces  of 
chalk ;  and  so  on.  A  limit,  though,  will/  be  reached  when 
a  piece  of  chalk  is  obtained  which  is  so  small  that  if  it  is 
separated  into  portions,  these  cease  to  have  the  properties 
of  chalk.  This  final  piece  of  chalk  is  called  a  "  molecule  " 
of  chalk.  So  molecules  of  other  substances  can  be  defined 
as  those  portions  of  the  substances  which  are  so  small 
that  if  they  are  separated  into  parts,  these  will  have  en- 
tirely different  properties.  Any  piece  of  matter  can,  then, 
be  regarded  as  made  up  of  molecules ;  and  experiments 
show  that  if  the  matter  is  homogeneous,  the  molecules  are 
identical  in  every  respect.  These  molecules  are,  in  general, 
groups  of  still  smaller  portions,  which,  so  far  as  we  now 
know,  cannot  be  subdivided.  In  certain  substances  not 
alone  are  the  molecules  identical,  but  the  molecular  sub- 
divisions also ;  and  such  a  body  is  said  to  be  "  elementary  " 
or  -an  "  element."  Thus  hydrogen  is  an  element ;  because 
not  alone  are  its  molecules  alike,  but  the  ultimate  portions 
of  the  molecule  are,  so  far  as  we  now  know,  identical. 
Similarly  oxygen  is  an  element.  Steam  is  not  an  element, 
because,  although  the  molecules  are  identical,  the  portions 
of  a  molecule  are  not  alike,  but  consist  of  two  portions 
like  the  final  portions  of  hydrogen  and  one  portion  like  the 
final  portions  of  oxygen.  Thus  steam  is  said  to  be  a 
"  compound  "  of  hydrogen  and  oxygen.  Of  course,  the  word 
"  final "  is  used  simply  to  mark  the  limit  reached  at  pres- 
ent ;  further  research  may  produce  still  further  subdivision. 
But  this  branch  of  science  which  considers  the  divisions 
of  molecules  and  their  rearrangement  into  compounds,  be- 
longs to  Chemistry.  For  the  purposes  of  Physics  it  is  suf- 
ficient to  regard  all  matter  as  groups  of  molecules,  the 
molecules  of  any  pure  substance  being  identical  in  all 


6  THEORY  OF  PHYSICS  [INT. 

respects ;  and  to  remember  that  portions  of  a  molecule 
have  properties  which  are  different  from  those  of  the 
molecule  itself. 

5.  Kinetic  Theory  of  Matter.     As  will  be  proved  later  on, 
there  is  conclusive  evidence  that  all  the  smallest  portions 
of  matter  are  in  motion.     The  material  body  itself  may 
not   move ;  but   the  molecules  and   the  portions    of   the 
molecules  are   in  constant   motion.     The    differences   be- 
tween   solids,   liquids,  and   gases  depend  largely  on    the 
freedom  which  the  molecules  have  for  motion.     In  a  gas 
the  molecules  can  move  with  almost  perfect  freedom.    They 
cannot  move  very  far  without  meeting   other  molecules 
or  the  walls  of  the  vessel,  when  their  motion  is  changed ; 
but,  owing  to  the  comparatively  large  spaces  between  mole- 
cules, there  is  great  freedom.     In  a  liquid,  molecules  can 
also  move  from  one  portion  to  another  with  comparative 
freedom ;  but  the  molecules  are  here  so  close  together  that' 
they  influence  each  other  greatly.     In  a  solid  the  mole- 
cules cannot  move  from  one  point  to  a  distant  one.     The 
molecules  of  a  solid  may  best  be  regarded  as  forming  a 
framework  or  stable  configuration,  which  is  constantly  in 
vibration,  owing  to  ther  vibrations  of  the  molecules  them- 
selves ;  that  is,  the  molecules  of  a  solid  do  not  leave  a 
definite  position,  but  simply  vibrate  around  it  or  through 
it.     The  passage,  then,  of  matter  from  the  state  of  a  solid 
into  that  of  a  liquid  and  then  into  that  of  a  gas  consists  in 
its  molecules  being  given  greater  and  greater  freedom  of 
motion.     This  view  of  the  properties  of  matter  is  often 
called  the  "  Kinetic  Theory." 

6.  Quantity  of  Matter.     To  compare   different   amounts 
of  matter,  any  one  of  the  general  properties  of  matter  may 
be  taken  as  the  basis  of  comparison.     Thus,  if  two  portions 
of  matter  have  the  same  inertia,  or  if  they  have  the  same 
weight  with  reference  to  the  earth  at  a  certain  point,  they 
may  be  defined  as  having  the  same  quantities  of  matter. 
The  quantity  of  matter  in  a  body  is  ordinarily  called  its 


8]  INTRODUCTION  7 

"  mass  ;  "  and  so  equal  masses  may  be  defined  in  either  of 
these  two  ways.  As  will  be  seen  later  on,  if  two  bodies 
have  the  same  inertia,  they  also  have  the  same  weight ;  so 
that  both  modes  of  measurement  of  masses  will  give  the 
same  results.  The  actual  methods  of  measurement  will 
be  described  in  full  further  on. 

If  any  number  of  substances  are  placed  inside  a  vessel 
so  made  that  no  matter  can  pass  in  or  out,  and  if  these 
substances  react  on  each  other  in  any  way,  or  if  there  are 
changes  of  any  kind,  it  may  be  proved  by  careful  experi- 
ments that  the  mass  is  the  same  at  all  times.  This  fact  is 
sometimes  called  the  "  Principle  of  the  Conservation  of 
Matter,"  and  it  further  asserts  that  the  amount  of  matter 
in  the  Universe  is  constant.  The  science  of  Chemistry  is 
based  upon  this  principle. 

7.  Motion.     Since,  then  in  any  natural  phenomenon  the 
amount  of  matter  is  not  altered,  the  only  change  possible, 
if  matter  is  involved,  is  a  change  in  the  position  of  the 
matter.     The  body  may  move  as  a  whole,  or  its  smaller 
portions  may  move;  but  in  all  cases  every  phenomenon 
of  matter  must  be  reduced  to  a  question  of  motion.     So 
to  "  explain "   every  phenomenon  is    the  aim   of  scientific 
research. 

Motion  involves  two  ideas,  space  and  time,  —  the  distance 
moved  over  and  the  time  taken.  Consequently,  the  funda- 
mental notions  of  every  material  phenomenon  are  mass, 
space,  and  time ;  and  in  describing  any  phenomenon  o^ 
stating  its  law  of  action,  certain  definite  amounts  of  each 
of  these  quantities  are  involved. 

8.  Fundamental  Units.     Before  an  amount  of  any  quan- 
tity can  be  expressed  mathematically,  a  unit,  or  standard, 
by  which  to  measure  it,  must  be  selected.     The  following 
physical  units  have  been  adopted  by  the  scientific  world. 

The  centimetre.  The  unit  of  length  is  called  the  "  centi- 
metre," and  it  is  the  one  hundredth  portion  of  the  length 
of  a  certain  metal  bar  called  the  metre,  which  is  kept  in 


8  THEORY  OF  PHYSICS  [INT. 

Paris,  the  length  of  the  bar  being  measured  when  it  is 
surrounded  by  melting  ice,  i.  e.  at  the  temperature  of 
0°  centigrade.  The  unit  of  area  is  the  square  centimetre  ; 
and  the  unit  of  volume  is  the  cubic  centimetre. 

When  this  bar  in  Paris  was  constructed,  it  was  designed 
to  have  a  length  equal  to  one  ten-millionth  of  the  distance 
from  the  equator  to  the  pole  of  the  earth  along  any  meri- 
dian ;  but  now  that  later  measurements  have  shown  that 
this  relation  between  the  bar  and  the  earth  is  not  exact, 
the  scientific  unit  of  length  refers  to  the  bar  itself,  and 
has  no  connection  with  the  size  or  shape  of  the  earth. 
Many  copies  of  the  standard  metre -bar  have  been  made ; 
and  although  none  are  exact,  their  errors  are  accurately 
known. 

The  gram.  The  unit  of  mass,  i.  e.  the  unit  amount 
of  matter,  is  called  the  "  gram ; "  and  it  is  the  one-thou- 
sandth portion  of  the  mass  of  a  certain  piece  of  metal 
called  the  kilogram,  which  is  kept  in  Paris.  Many  copies 
of  the  kilogram  have  been  made ;  and  although  none  of 
them  are  exactly  correct,  yet  the  differences  between  them 
and  the  standard  have  been  most  carefully  measured. 

When  this  standard  mass  in  Paris  was  made,  it  was 
designed  to  be  of  such  an  amount  that  the  mass  of  one- 
thousandth  of  it  should  be  the  same  as  that  of  one  cubic 
centimetre  of  distilled  water  when  it  is  densest,  i.  e.  at 
the  temperature  of  4°  centigrade.  This  relation  has  been 
shown  to  be  not  perfectly  exact ;  and  the  standard  of 
mass,  then,  is  the  gram  itself,  not  the  cubic  centimetre  of 
water.  But  for  all  ordinary  practical  purposes  we  may 
regard  the  mass  of  one  cubic  centimetre  of  water  at  4° 
centigrade  as  being  one  gram. 

The  "  density  "  of  any  homogeneous  substance  is  defined 
as  the  number  of  grams  in  one  cubic  centimetre  of  the 
substance,  and,  in  general,  the  density  of  any  homogeneous 
portion  of  the  substance  is  the  mass  of  that  portion 
divided  by  its  volume.  Thus  the  density  of  water  at  4° 


9]  INTRODUCTION  9 

centigrade  is  one  ;  and  methods  will  be  described  later  on 
for  the  measurement  of  the  densities  of  all  other  bodies. 

The  second.  The  unit  of  time  is  called  the  "  second ; " 
and  it  is  the  second  with  reference  to  the  so-called  "  mean 
solar  day."  That  is,  the  second  which  is  the  scientific 
unit  of  time  is  the  g-gitTo  portion  of  the  average  length  of 
a  solar  day  for  one  year. 

The  solar  day  is  the  interval  of  time  which  elapses 
between  two  successive  transits  of  the  sun  across  the 
meridian  of  the  earth  at  the  point  of  observation  ;  and  this 
interval  is  not  the  same  at  all  times  of  the  year.  The 
average  length  of  the  intervals  for  one  year  is,  however, 
constant  from  year  to  year,  so  far  as  we  now  know. 

These  units  are,  of  course,  perfectly  arbitrary,  and  are 
not  basea  on  quantities  which  are  in  themselves  constant. 
But  they  are  most  convenient,  are  simply  denned  and  con- 
nected, and  are  in  universal  use.  The  system  of  measure- 
ment based  upon  them  is  called  the  "C.  G.  S.  System," 
from  the  initial  letters  of  the  three  standards ;  and  the 
symbols  ordinarily  used  are  "  cm.,"  "  g.,"  "  sec." 

Methods  for  the  measurement  of  lengths,  masses,  and 
intervals  of  time  are  fully  described  in  laboratory  manuals, 
and  so  need  not  be  given  here. 

9.  Mechanics.  As  has  been  said,  all  phenomena  depend- 
ing upon  matter  involve  simply  the  idea  of  motion  of 
matter.  And  so  the  study  of  Physics  must  begin  with 
a  discussion  of  the  different  possible  kinds  of  motion,  o<" 
the  properties  of  matter  in  motion,  and  of  the  conditions 
under  which  these  motions  may  occur.  This  discussion 
forms  what  is  called  the  science  of  "  Mechanics." 


THEORY  OF   PHYSICS 
BOOK  I 

MECHANICS   AND   PROPEKTIES   OF    MATTEE 


CHAPTER  I 

KINEMATICS 

KINEMATICS  is  that  branch  of  Mechanics  which  is  con- 
cerned with  the  different  possible  kinds  of  motion,  and 
with  a  discussion  of  their  distinguishing  features. 

10.  Motion  in  General.  If  any  body,  such  as  a  book  or 
a  stone,  is  tossed  in  the  air  at  random,  it  is  at  once  evi- 
dent that  there  are  two  distinct  motions :  the  body  moves 
as  a  whole;  and  at  the  same  time  it  is  twisting.  It  is 
possible,  of  course,  to  have  motion  such  that  one  only  of 
these  types  occurs.  Thus,  if  a  book  be  raised  from  a  table 
so  that  all  lines  which  could  be  imagined  drawn  through 
the  book  remain  parallel  to  themselves,  the  book  simply 
moves  as  a  whole,  it  does  not  twist.  This  motion  is  called 
pure  "  translation  ;"  and  it  may  be  denned  as  such  motion 
that  all  the  points  of  the  body  move  through  equal  parallel 
paths.  Again,  if  some  point  of  a  body  is  fixed,  the  body 
cannot  move  as  a  whole,  it  can  only  spin  or  turn.  An 
illustration  of  this  is  a  spinning  top  whose  point  does  not 
move,  or  a  swinging  door.  In  the  case  of  the  door,  the 
points  on  the  line  through  the  hinges  are  at  rest,  while  the 
other  points  of  the  door  are  all  describing  circles  around 
this  line.  In  the  case  of  a  spinning  top,  the  geometrical 
axis  may  be  considered  at  rest  at  any  one  instant,  and  all 
the  points  of  the  top  are  describing  circles  around  it.  If 
the  axis  of  the  top  is  inclined  to  the  vertical,  it  does  not 
remain  at  rest,  but  changes  its  direction.  Such  motion  as 


14  THEORY  OF  PHYSICS  [CH.  I 

that  of  a  door  or  a  top  is  called  "  rotation ; "  and  it  may 
be  defined  as  motion  where  at  least  one  point  of  the  body 
is  at  rest,  while  the  others  are  moving  at  any  instant  in 
circular  paths  around  a  line,  called  the  "axis,"  which 
passes  through  the  fixed  point.  The  axis  may  be  fixed, 
as  in  an  ordinary  door  or  fly-wheel ;  or  it  may  be  changing 
its  direction,  as  in  a  spinning-top  when  the  axis  of  figure  is 
not  vertical.  The  most  general  type  of  motion  is  a  com- 
bination of  translation  and  rotation  such  as  the  motion  of 
a  nut  on  a  screw. 

It  should  be  clearly  understood  that  all  motion  is  purely 
relative.  If  it  is  said  that  a  stone  falls  in  a  straight  line, 
it  is  meant  that  the  path  of 'the  stone  with  reference  to 
the  earth  is  a  straight  line.  The  actual  path  of  the  stone 
in  space  will  be  entirely  different  from  a  straight  line, 
owing  to  the  motion  of  the  earth  itself,  of  the  sun,  etc. 
If  in  a  railway-carriage  running  north,  a  passenger  walks 
directly  across  the  aisle  from  the  east  side  to  the  west,  his 
motion  with  reference  to  the  carriage  is  west ;  with  refer- 
ence to  the  earth,  it  is  northwest ;  with  reference  to  the 
sun,  it  depends  upon  the  hour  and  day ;  etc.  It  is  impos- 
sible to  conceive  what  the  true  motion  is,  because  the 
human  mind  cannot  imagine  any  point  absolutely  at  rest. 
Similarly,  in  motion  of  rotation,  the  actual  true  motion 
cannot  be  conceived;  and  only  the  relative  motion  is 
studied.  Ordinarily  in  Physics  all  motions  are  referred 
to  the  earth  as  at  rest. 

11.  Translation.     In  translation  all  the  points  of  a  body 
move  through  equal   and  parallel  paths ;    so  that,  if  the 
change  of  position  of  any  one  point  of  the  body  is  known, 
the  motion  of  the  whole  body  is  also  known.      To  de- 
scribe and  understand  translation  completely,  then,  the 
motion  of  a  point  must  be  studied;  that  is,  its  rate  and 
direction  of  motion  at  successive  instants  must  be  known. 

12.  The  change  of  position  of  a  point  is  called  the  "  dis- 
placement ; "  and  the  rate  of  this  change  with  respect  to 


12]  KINEMATICS  15 

the  time  is  called  the  "linear  velocity."  If  this  change 
is  uniform,  e.  g.  a  railway  train  running  at  a  constant 
rate,  the  velocity  is  the  displacement  in  one  second,  and 
is  the  same  for  successive  instants.  If,  however,  the 
change  of  position  is  not  uniform,  e.  g.  a  train  slowing 
up,  the  velocity  at  any  instant  is  the  displacement  which 
would  occur  in  one  second  if  the  rate  of  motion  remained 
for  one  second  the  same  as  it  is  at  that  instant.  Thus, 
when  a  train  is  said  to  be  going  at  the  rate  of  sixty  miles 
an  hour,  it  is  not  implied  that  the  train  will  actually  go 
sixty  miles  in  the  following  hour,  but  that  it  would  do 
so  if  its  rate  were  to  remain  unchanged. 

If  a  point  is  displaced,  two  ideas  are  involved,  —  distance 
and  direction.  Thus,  if  the  point  moves  from  P  to  Q,  the 
displacement  is  the  line  PQ,  which 
has  a  certain  length,  and  the  direc- 
tion from  P  to  Q.  (This  is  often  in- 
dicated by  drawing  an  arrow-head 
on  the  line  in  the  direction  of  the 
motion.)  The  line  PQ  is  of  course 
equal  to  —  Q  P,  because  if  the  point 
moves  from  P  to  Q,  and  then  back  to 
A,  the  displacement  is  zero.  Hence 

~p~Q  +  ~qp=.  0.  This  idea  of  lines  having  direction  as  well 
as  length  should  be  familiar  to  all  from  a  study  of  trigo- 
nometry. Since,  then,  displacement  implies  direction  as 
well  as  distance,  linear  velocity  must  mean  rate  of  mo- 
tion in  a  particular  direction.  Thus,  a  train  going  north 
at  the  rate  of  1000  centimetres  per  second  is  said  to  have 
the  velocity  "1000  north;"  while  a  train  going  south  at 
the  same  rate  has  a  velocity  "1000  south;"  and  the  two 
velocities  are  equal  and  opposite,  just  as  5  and  —  5  are 
equal  and  opposite,  or  as  the  line  P  Q  is  equal  and  oppo- 
site to  the  line  Q  P.  To  express  completely,  then,  the 
linear  velocity  at  any  instant,  two  quantities  are  necessary, 
—  a  number,  to  give  the  rate  of  motion,  and  a  direction. 


16  THEORY  OF  PHYSICS  [CH.  I 

This  number,  which  gives  the  rate  of  motion,  without 
any  statement  of  direction,  is  called  the  "linear  speed." 
If  the  speed  does  not  change,  it  represents  the  act- 
ual distance  in  centimetres  which  is  passed  over  in  one 
second ;  and  the  distance  traversed  in  t  seconds  is  the 
product  of  t  and  the  speed.  The  speed  at  any  instant 
is  measured  by  determining  how  many  centimetres  are 
passed  over  in  a  certain  time,  and  then  dividing  the  dis- 
tance by  the  time.  Experimental  methods  are  taught 
in  laboratories. 

Since  a  straight  line  has  the  same  two  properties  as  a 
velocity,  —  a  number,  giving  its  magnitude,  and  a  definite 
f  direction, — it  can  be  used  to  represent  the  lin- 
ear velocity  at  any  instant.  Thus  let  the  velo- 
city at  any  instant  be  "  10  north ; "  a  straight 
line  drawn  from  A  to  B  in  a  northern  direction, 
and  having  a  length  of  ten  arbitrary  units,  will 
completely  represent  the  velocity.  The  length 
of  the  line  gives  the  speed,  and  its  direction 
must  be  the  same  as  that  of  the  displacement 
at  that  instant.  Any  other  line,  A1  B1 ',  parallel 
"FIG  2  ^  to  A  B  and  of  the  same  length,  would  do  just 
as  well,  because  being  parallel  they  have  the 
same  direction,  and  they  are  made  of  the  same  length. 
As  an  illustration  of  this  mode  of  representation  of  linear 
velocities,  consider  the  motion  of  a  point  in  a  circle,  the 
speed  being  constant,  e.  g.  a  stone  whirled  in  a  sling, 
in  a  horizontal  plane.  When  the  point  is  at  P,  its  direc- 
tion of  motion  is  along  the  tangent  to  the  circle  at 
P,  i.  e.  perpendicular  to  the  radius  OP.  Hence,  if  the 
speed  is  s,  the  velocity  at  P  may  be  represented  by  a 
line,  A  B,  perpendicular  to  0  P,  and  having  a  length 
equal  to  s  arbitrary  units.  Similarly,  at  Q  the  velocity 
may  be  represented  by  a  line,  A1  Br,  perpendicular  to  the 
radius  0  Q,  and  having  a  length  s,  because  the  speed  re- 
mains constant. 


13] 


KINEMATICS 


17 


FIG.  3. 

13.  It  often  happens  that  a  body's  actual  displacement 
is  due  to  two  or  more  causes,  which  by  themselves  indi- 
vidually would  have  produced  different  displacements. 
The  actual  displacement  may,  then,  be  regarded  as  the 
sum  of  the  separate  ones.  Thus,  when  a  man  walks  across 
a  railway  carriage  which  is  in  motion,  his  actual  displace- 
ment with  reference  to  the  earth  is  made  up  of  two  parts,  — 
his  motion  across  the  carriage,  which 
may  be  represented  by  the  line  A  B, 
and  the  motion  of  the  carriage  with 
reference  to  the  earth,  which  may  be 
represented  by  the  line  B  C.  So  that, 
starting  from  the  point  A,  the  man 
reaches  the  point  C  with  reference  to 
the  earth.  If  these  distances  are  all  z 
passed  over  in  one  second,  the  lines  A 
A  B,  B  C,  and  A  C  represent  veloci- 
ties ;  and  it  is  thus  proved  that  if  a  point  is  subjected  to 
two  velocities,  A  B  and  B  0,  the  actual  velocity  will  be 
given,  in  direction  and  amount,  by  the  line  A  0.  Simi- 
larly, consider  a  man  rowing  a  boat  across  a  river  which 
has  a  strong  current.  Let  A  B  be  the  velocity  given  the 
boat  by  the  oars,  and  B  C  be  that  given  it  by  the  current ; 
then  the  velocity  of  the  boat  at  any  instant  will  be  rep- 
resented by  the  line  A  C,  which  completes  the  triangle, 


FIG.  4. 


18 


THEORY  OF  PHYSICS 


[CH.  I 


two  of  whose  sides  are  A  B  and  B  C.  This  process  is 
sometimes  spoken  of  as  "  composition  of  velocities,"  or 
"  geometrical  addition."  In  adding 
lines  in  this  way,  care  must  be 
taken  to  join  two  ends  which  are 
such  that  the  arrows  indicate  con- 
tinuous motion  through  the  junc- 
tion. The  numerical  value  of  A  C, 
or  the  actual  speed,  is  found,  by 
means  of  trigonometry,  from  a 
knowledge  of  the  lengths  of  the 
other  two  sides  and  the  angle  be- 
tween them.  Using  the  symbol 
A  £  to  mean  the  line  A  to  B,  not 
its  mere  length,  the  above  facts 
may  be  written 

A   n ~2    z?    I     p  ri . 

-fl.  L/  —  .^  -D  -t-  .Z5  U, 

and  from  this  it  follows  that 


FIG.  5. 


AC  -AB  =  B  C. 
The  difference  between  two  ve- 
locities is  thus  found  by  drawing 

the  lines  which  represent  them,  both  away  from  the  same 
point,  and  then  constructing  the  third  side  of  the  triangle. 

If  a  point  has  more  than  two  velocities  given  it  simul- 
taneously, the  actual  velocity  may  be  found  by  adding  two, 
then  adding  the  third  to  their  sum,  and  so  on ;  or,  what  is 
the  same  thing,  join  all  the  lines  which  represent  the 
velocities,  in  such  a  way  that  the  arrows  point  in  a  con- 
tinuous direction,  and  then  draw  a  line  to  complete  the 
polygon.  This  will  be  the  actual  velocity. 

14.  If  the  motion  of  translation  is  uniform,  it  is  com- 
pletely denned  by  a  knowledge  of  the  linear  velocity ;  but 
in  all  actual  cases  the  velocity  is  changing,  and  so  a 
knowledge  of  the  nature  and  amount  of  this  change  is 
required.  The  name  "  linear  acceleration  "  is  given  to  the 


15] 


KINEMATICS 


19 


rate  of  change  of  the  linear  velocity  with  reference  to  the 
time ;  and  it  may  be  uniform  or  not.  If  the  change  is 
uniform,  the  acceleration  is  the  change  in  the  velocity  in 
one  second ;  if  it  is  not  uniform,  the  acceleration  at  any 
instant  is  the  change  in  the  velocity  which  would  take 
place  in  one  second  if  the  rate  of  change  did  not  vary  in 
that  time.  Since  velocity  has  two  essential  properties, 
direction  and  amount  (i.  e.  speed),  it  may  be  changed  in 
two  ways,  either  in  direction  or  in  speed.  Thus  there  are 
two  types  of  linear  acceleration :  1.  Where  the  direction 
is  unchanged,  but  the  speed  is  altered,  e.  g.  a  falling  body ; 
2.  Where  the  speed  is  unchanged,  but  the  direction  is 
altered,  e.  g.  a  stone  whirled  in  a  sling  in  a  horizontal 
plane.  In  the  general  case,  both  speed  and  direction  are 
changing. 

An  acceleration  can  also  be  repre- 
sented by  a  straight  line,  because,  be- 
ing the  rate  of  change  of  velocity,  it 
is  the  difference  between  two  veloci- 
ties. It  has  both  a  definite  direction 
and  a  definite  amount. 

15.  Accelerations  can  be  added,  then, 
or  compounded.  Thus,  let  a  point  be 
subject  to  two  accelerations,  A  B  and 
B  C\  the  actual  acceleration  will  then 
be  the  line  A  C,  which  is  the  geomet- 
rical sum  of  A  B  and  B  C.  Looking 
at  this  theorem  in  the  reverse  way,  it 
may  be  said  that  any  actual  accelera- 
tion may  be  regarded  as  the  sum  of 
any  two  accelerations  which  when 
added  geometrically  equal  it.  There 
is  thus  an  indefinite  number  of  ways 
in  which  an  acceleration  may  be  FIG.  6. 

broken  up,  or  "  resolved,"  into  parts. 
In  the  figure  the  acceleration  A  C  is  regarded  as  equiva- 


20 


THEORY   OF  PHYSICS 


[CH.  I 


BC  in  the  direction  B  to 

C 


B 


lent  to  an  acceleration  A  B  in  the  direction  A  to  B,  and 

C.  But  this  last  acceleration 
also  produces  a  certain  effect 
in  the  direction  A  to  B.  So 
that,  if  A  C  is  to  be  resolved 
into  two  mutually  indepen- 
dent accelerations  A  B  and 
B  C,  these  last  must  be  at 
right  angles  to  each  other.  In 
this  case  the  total  effect  in  the  direction  A  to  B  of  the 
acceleration  A  C  is  given  by  the  line  A  B ;  and  the  total 
effect  in  the  direction  B  to  C  is  given  by  the  line  B  C.  For 
this  reason  the  acceleration  A  B  is  called  the  resolved  por- 
tion of  the  acceleration  A  C  in  the  direction  A  to  B.  If  6 
is  the  angle  between  the  lines  A  to  C  and  A  to  B, 


FIG.  7. 


B~C=i  A~C  sin  0. 

An  illustration  of  this  is  afforded  by  a  body  sliding  down 
an  inclined  plane.  Let  the  angle  between  the  plane  and 
the  horizontal  plane  be 
<f>,  and  let  the  body  be 
at  A  at  any  instant.  If 
the  plane  were  not  there, 
the  body  would  experi- 
ence an  acceleration  A  C 
vertically  downwards. 
A  C  can  be  regarded  as 
producing  the  accelera-  FIG.  8, 

tions  A  B  and  B  C  per- 
pendicular and  parallel  to  the  plane  respectively ;  but 
now,  if  the  body  rests  on  the  plane,  the  acceleration 
A  B  perpendicular  to  the  plane  is  prevented,  and  the 
actual  acceleration  will  be  B~C  parallel  to  the  plane.  Fur- 
ther WC-~AC  sin  (BAC)  =  A~C  sin  <£.  This  deduction 


16] 


KINEMATICS 


21 


may  be  verified  by  experiment,  as  both  A  0  and  B  C  can 
be  measured. 

In  an  exactly  similar  way  a  velocity  may  be  resolved 
into  components  in  directions  at  right  angles  to  each 
other. 

16.  Rotation.  In  rotation,  at  least  one  point  of  a  body  is 
at  rest,  and  all  the  other  points  are  at  any  instant  moving 
in  circles  whose  centres  all  lie  on  a  line  which  passes 
through  the  fixed  point.  Consider  any  plane  section  of  the 
body  perpendicular  to  the  axis  ; 
draw  lines  A  A  and  B  B' \  in 
this  plane  of  the  body,  also  draw 
a  line,  PP' ',  lying  in  this  plane, 
but  fixed  in  space.  Let  0  be  the 
point  where  the  plane  cuts  the 
axis  ;  then,  if  the  axis  is  fixed, 
the  lines  A  A'  and  B  B'  will 
make  certain  angles  with  the 
fixed  line  PP',  which  will  change 
with  time  as  the  body  rotates. 
But  the  change  in  the  angle 

made  between  A  A'  and  P  P'  equals  the  change  in  the 
angle  made  between  B  B'  and  P  P  because  A  A'  and  B  B' 
have  a  constant  angle  between  them.  This  is  equivalent 
to  saying  that  all  lines,  A  A',  B  B1,  etc.,  fixed  in  the  body 
turn  through  equal  angles  with  reference  to  any  line,  PP', 
fixed  in  space,  these  lines  all  being  drawn  in  a  plane  per- 
pendicular to  the  axis  of  rotation.  Consequently,  just  as 
in  motion  of  translation  all  points  of  the  body  move 
through  the  same  distances  in  the  same  direction,  so  in 
motion  of  rotation  all  lines  in  a  plane  perpendicular  to  the 
direction  of  the  axis  have  the  same  angular  motion  around 
that  axis.  The  angle  between  the  line  A  A'  fixed  in  the 
body  and  PP'  fixed  in  space  determines  the  position  of  the 
body ;  and  the  rate  of  change  of  this  angle  with  respect  to 
the  time  is  called  the  "angular  velocity."  It  is  evident 


FIG.  9. 


22  THEORY  OF  PHYSICS  [CH.  I 

that  two  quantities  are  necessary  for  its  complete  definition ; 
the  direction  of  the  axis  so  that  a  plane  may  be  drawn  per- 
pendicular to  it,  and  the  numerical  value  of  the  rate  of 
angular  motion,  i.e.  the  "angular  speed."  Consequently 
an  angular  velocity  may  be  represented  by  a  straight  line. 
Thus  A  B  may  represent  the  turning 
of  a  body  whose  axis  at  the  instant 
has  the  direction  A  to  B,  and  whose 
rate  of  turning  at  the  instant  equals 
the  numerical  length  of  A  B.  A  line 
B  A  would  represent  a  body  turning 
with  the  same  speed  around  the  same 
FlG  10  axis,  only  in  the  opposite  direction. 

And  just  as  linear  volocities  may  be 

added,  so  may  angular  ones,  if  the  axes  of  rotation  of  the 
different  motions  meet  in  a  point. 

In  translation  either  the  direction  or  the  speed  may 
change,  and  thus  produce  linear  acceleration ;  so,  in  rota- 
tion, either  the  direction  of  the  axis  may  change,  as  in  the 
case  of  a  spinning  top  whose  axis  of  figure  is  not  vertical, 
or  the  angular  speed  may  change,  the  direction  of  the  axis 
remaining  unaltered,  as  in  the  case  of  a  barrel  rolling  down 
an  inclined  plane.  Thus  there  are  two  types  of  angular 
acceleration,  and  the  general  case  is  a  combination  of  these 
two ;  that  is,  both  the  direction  of  the  axis  and  the  angular 
speed  around  the  axis  change.  Since  angular  acceleration 
is  the  rate  of  change  of  angular  velocity,  it  can  be  repre- 
sented by  a  straight  line ;  and  angular  accelerations  can 
be  added  or  resolved  into  components  around  particular 
axes,  just  as  linear  accelerations  can  be  added  or  resolved 
into  components  in  particular  directions. 

17.  Numerical  Value  of  an  Angle.  It  may  be  well,  at  this 
point,  to  recall  the  scientific  definition  of  an  angle  and  its 
numerical  value.  An  angle  between  two  lines  lying  in  the 
same  plane  is  the  difference  in  direction  between  them; 
and  its  numerical  value  is  defined  in  this  way :  prolong 


18]  KINEMATICS  23 

the   lines   until  they  intersect,  then  with  any  radius,  r, 

describe  a  circular  arc  around  this  point  of  intersection 

as    a    centre,    having 

the  arc  terminated  by 

the    two    lines.      Let 

the  length  of  the  arc 

be    a,    then    the    nu-  ^'" 

nierical   value  of   the    ^_^ 

angle   is    a  jr.      This    p  FIG  n  p 

value   is   independent 

of  the  length  of  the  radius  which  is  taken,  as  appears 
from  ordinary  geometry. 

As  illustrations :  a  right  angle  has  the  numerical  value 

2  TT  r  ,         TT 

— ] —  /  T  —  TT  »   one  entlre  turn   corresponds  to  an  entire 

4  _ 

circumference,  and  has  the  numerical  value  2  TT  r /  r  =  2  IT. 

SPECIAL   CASES 
I.   TRANSLATION 

18.    1.  Motion  in  a  Straight  Line ;  Constant  Acceleration. 

The  problem  is  to  determine  the  velocity  and  the  dis- 
tance passed  over  at  the  end  of  a  certain  time. 

Let  s0  be  the  initial  speed,  that  is,  the  speed  at  the 
instant  from  which  time  is  counted;  a,  the  acceleration, 
which  in  this  case  is  a  known  constant  quantity ;  t,  the 
number  of  seconds  at  the  end  of  which  it  is  desired  to 
know  the  velocity  and  the  distance  traversed.  Since  in 
this  problem  the  motion  is  in  a  straight  line,  the  accelera- 
tion is  the  numerical  change  in  the  speed.  Hence  at  the 
end  of  t  seconds  the  speed  will  be 

s  =  s0  +  a  t (1) 

Since  the  speed  is  changing  uniformly,  the  mean  speed 
during  the  t  seconds  is  -      — ,  the  numerical  average  of 


24  THEORY  OF  PHYSICS  [CH.  I 

the  initial  and  final  values.  This  is  the  average  distance 
passed  over  in  one  second;  and  so  the  entire  distance 
traversed  in  t  seconds  is 

gp  +  8 

-*-• 

Substituting  for  s  its  value  from  (1), 

x  =  sQt  +  ±at2 (2) 

Eliminating  t  from  (1)  and  (2), 

s*-s*  =  2ax    .     ...     .     .     (3) 

Conversely,  if  by  observation  it  is  found  that  the  motion 

s2  —  s02 

is  such  that  x  —  — ,  where  c  has  a  constant  numeri- 
cs c 

cal  value  for  all  values  of  x,  then  it  is  known  that  the 
acceleration   is    constant,   and   c   is   its    numerical   value. 
This  is  a  method  of  measuring  a  constant  acceleration. 
As  illustrations  of  these  formulae,  consider  two  examples. 
a.    A  body  falls  from  rest  towards  the  earth.     Experi- 
ments prove  that  falling  bodies  have  a  constant  accelera- 
tion towards  the  earth  of  about  980.     Hence,  since  s0  =  0 
and  a  =-.  980, 

s  =  980 1 ; 
x  =  I  -  980  .  t2 
s2  =  2  •  980  x. 

If  t  =  1, '  s  =  980,     x  =  490, 

if  t  =  2,   s  =  1960,   x  =  1960. 

&.  A  body  is  thrown  vertically  upwards  with  an  initial 
velocity  4900.  How  far  will  it  go  before  it  comes  to  rest  ? 
s0  =  4900 ;  a  =  —  980,  since  in  this  case  the  acceleration 
is  opposite  to  the  speed;  also  s  =  0,  since  the  final  velocity 
is  to  be  0. 
Hence  0  =  4900  —  980  t ; 

a;  =  4900* -490  £2; 
0  -  49002  =  -  2  •  980  •  x. 
Hence  t  =  5  ;   x  =  12250. 


19] 


KINEMATICS 


25 


\ 


19.    2.  Motion  in  a  Circle  ;  Constant  Speed. 

The  problem  is,  knowing  the  speed,  to  determine  the 
acceleration,  —  both  its  direction  and  its  amount.  Let  the 
motion  be  in  a  circle  of 
radius,  r,  around  a  cen- 
tre, 0 ;  and  let  P  and  Q 
be  two  points  of  the  cir- 
cle, which  are  so  near 
that  they  may  be  re- 
garded as  consecutive ; 
and  let  it  take  the  ex-  FIG.  12. 

tremely  small  interval  of 

time  t  seconds  for  the  point  to  move  from  P  to  Q  (t  may 
be  1  millionth,  or  even  smaller).  Then,  if  s  is  the  speed, 
P~Q  =  s  '  t  because  s  is  the  distance  passed  over  in  one 
second. 

The  velocity  at  the  point  P  may  be  represented  by  the 
line  0  A,  whose  length  is  s,  and  which  is  perpendicular  to 
0~P.  Similarly,  the  velocity  at  Q  is  given  by  the  line  0  B, 
whose  length  is  s,  and  which  is  perpendicular_to  0  Q. 
Hence  the  change  in  the  velocity  in  t  seconds  is  OB—  0  A, 
that  is,  the  line  A  B.  But  a,  the  acceleration,  is  the 
change  in  the  velocity  in  one  second;  hence 

~TB  =  a  .  t 

Hence,  eliminating  t  from  the  two  equations, 

a       Til 


But  since  the  angles  A  OB  and  a  0 1  are  equal, 

0~A       s 


Hence 

and  so 


AB_  = = 

TQ~  OT~ 

a  _  s 

a  =  s2  /  r 


(4) 


26  THEORY  OF  PHYSICS  [CH.  I 

This  is  the  numerical  value  of  the  acceleration  when  the 
point  is  at  P ;  its  direction  is  given  by  the  line  A  B  when 
the  time  t  becomes  infinitely  small.  In  the  limit,  0  Q 
coincides  with  OP;  and  the  line  A  B  is  parallel  to  them. 
That  is,  the  acceleration  at  the  point  P  has  for  its  numer- 
ical value  s2/r,  and  is  directed  toward  the  centre  of  the 
circle.  If  this  acceleration  ceased  being  applied,  the  point 
would  move  off  in  a  tangent  to  the  circle  at  that  instant. 
This  explains  the  breaking  of  fly-wheels  when  the  speed 
becomes  too  great,  and  also  the  tendency  of  all  matter  to 
place  itself  as  far  as  possible  from  the  centre  when  it  is 
whirled  around  an  axis. 

As  the  point  moves  around  the  circle,  the  radius  joining 
it  to  the  centre  turns  through  a  constantly  increasing 
angle  with  reference  to  any  fixed  line.  The  angular  speed 
of  this  line  is  the  angle  turned  through  in  one  second,  but 
an  angle  has  for  its  numerical  value  the  ratio  of  the  arc  to 
the  radius.  So  that,  choosing  the  radius  as  that  of  the 
circle  r,  the  arc  passed  over  in  one  second  is  the  linear 
speed  s.  Hence,  calling  the  angular  speed  CD, 

co  —  s  /  r (5) 

And  substituting  for  s  its  value  in  terms  of  co,  (4)  becomes 

a  =  r  co2 (6) 

Call  n  the  number  of  complete  turns  which  the  point 
makes  around  the  circle  in  one  second. 

s  =  n  •  2  TT  T     and     co  =  n  •  2  TT. 

Hence  a  =  4  -rr2  n2  r (7) 

The  period  of  time  for  one  complete  revolution  around  the 
circle  is  -  ;  that  is,  2  TT  r  /  s  or  2  TT  /  co ;  and  calling  its 

71/ 

value  T,  the  formula  for  the  acceleration  becomes,  on  its 
substitution, 

a  =  ±TT*r  /  T2 (8) 


21] 


KINEMATICS 


27 


20.  Comparing  these  two  special  cases,  1  and  2,  it  is 
seen  that  the  speed  is  changed  by  applying  an  acceleration 
in  the  direction  of  motion ;  while  the  direction  is  changed 
by  applying  an  acceleration  at  right  angles  to  the  direction 
of    motion.     If    an   acceleration   is   applied   obliquely  to 
the  direction   of   motion,  it   may  be   considered   replaced 
by  its  two  components  parallel  and  perpendicular  to  this 
direction  ;  and  consequently  both  the  speed  and  direction 
will  be  changed. 

21.  3.    Harmonic  Motion.     This  is  a  motion  which  may 
be  thus  described :  Let  a  point,  P,  move  around  a  circle  of 
radius,  r,  with  a  constant 

speed,  and  take  at  consecu- 
tive instants  the  projection 
of  this  point  on  any  diame- 
ter of  the  circle.  Then, 
as  this  projected  point,  Q, 
moves  backward  and  for- 
ward along  the  diameter,  it 
has  a  certain  kind  of  vibra- 
tory motion,  which  is  called 
"  harmonic."  The  great  im- 
portance of  this  kind  of 
motion  is  due  to  the  fact 
that  it  occurs  so  often  in 

nature,  and  that  all  instruments  for  measuring  time  accu- 
rately have  motions  like  it. 

The  point  Q  has  a  certain  acceleration  along  the  diam- 
eter, and  the  problem  is  to  determine  the  connection 
between  this  acceleration  and  the  period  of  one  complete 
vibration  of  Q. 

The  acceleration  of  Q  towards  0  along  the  diameter  is 
the  same  as  the  component  of  the  acceleration  of  P  paral- 
lel to  the  diameter,  because  Q  is  the  projection  of  P  on  it. 
The  acceleration  of  P  towards  0  is  by  (6)  r  &>2  if  w  is  the 
constant  angular  speed  of  P.  Hence  the  component  of 


FIG.  13. 


28  THEORY   OF  PHYSICS  [CH.  I 

this  acceleration  parallel  to  the  diameter  is  r  tw2  •  cos 
(P  0  Q),  and  this  is  also  the  acceleration  of  Q  towards  0 
along  the  diameter.  Call  the  line  0  Q  x,  where  x,  of  course, 
is  a  variable  quantity,  its  greatest  value  being  +  r  and  its 
least  —  r.  Then  cos  (P  0  Q)  =  x/r,  and  the  acceleration  of 
Q  towards  0  may  be  written 

r  co2  x  IT,     or     x  o>2 ; 

i.  e.,  at  any  point  the  acceleration  equals  a  constant  times  x. 
The  period  of  a  complete  vibration  of  Q  is  the  same, 
obviously,  as   the   period   of  revolution    of  P  around  the 
circle.     Hence 


T  —  2  TT  /  (o=  2  TT  /  A/coefficient  of  x  in  the  acceleration.  (9) 

Therefore,  simple  harmonic  motion  is  such  motion  that  at 
any  instant  the  acceleration  is  towards  the  centre  of  the 
path,  and  has  for  its  value  a  constant  times  the  distance 
of  the  point  from  the  centre  ;  and,  further,  the  period  of 
vibration  is  equal  to  2  TT  divided  by  the  square  root  of  this 
constant.  The  length  of  the  path  (i.  e.  the  diameter  of  the 
circle)  is  called  the  "  amplitude  "  of  the  vibration.  Illus- 
trations of  harmonic  motion  are  afforded  by  the  vibrations 
of  pendulums,  tuning-forks,  stretched  strings,  etc. 

II.      EOTATION 

22.  1.  Motion  around  an  Axis  whose  Direction  does  not 
change ;  Angular  Acceleration  Constant. 

This  is  perfectly  analogous  to  Case  1,  in  Translation. 
Let  o>0  be  the  initial  angular  speed,  and  a  the  angular  ac- 
celeration ;  then,  at  the  end  of  t  seconds,  the  speed  o>,  and 
the  angle  turned  through,  0,  will  be 

to  =  o)0  +  a  t     . (10) 

e  =  2l±2-t  =  »tt+lafit     .     .     (11) 
and  a)2  -  W2  =  2  a  6 (12) 


24]  KINEMATICS  29 

As  illustrations  of  this  motion,  consider  a  cylinder  rolling 
down  an  inclined  plane  and  a  fly-wheel  being  stopped  by 
a  brake  applied  at  its  rim. 

23,  2.  Motion  of  a  Spinning  Top  whose  Axis  of  Figure  is 
not  Vertical.     The  axis  changes  its  direction  at  a  uniform 
rate,  and  the  angular  speed  around  the  axis  is  constant. 

This  is  perfectly  analogous  to  Case  2  in  Translation.  In 
that  problem  it  was  shown  that  in  order  to  make  the  point 
change  its  direction  at  a  uniform  rate  (i.  e.  to  make  it 
move  in  a  circle)  the  speed  remaining  constant,  it  is  neces- 
sary to  apply  continuously  to  the  point  a  constant  accelera- 
tion whose  direction  is  perpendicular  to  the  direction  of 
motion.  So  in  this  problem  of  rotation,  in  order  to  make 
the  axis  change  its  direction  at  a  uniform  rate,  the  angular 
speed  remaining  constant,  it  is  necessary  to  apply  a  con- 
stant angular  acceleration  around  an  axis  which  is  perpen- 
dicular to  the  axis  of  rotation.  In  the  case  of  a  spinning 
top  there  is  a  constant  tendency  to  fall  over,  i.  e.  to  turn 
around  a  horizontal  axis ;  and  this  axis  is  perpendicular  to 
the  axis  of  rotation,  hence  this  last  axis  changes  its  direc- 
tion at  a  uniform  rate. 

24.  In  general,  then,  if   a   body  is   spinning  with  the 
angular  velocity  A  B,  the  angular  speed  can  be  changed 
by  applying  an  angular   acceleration 

around  the  same  axis  ;  but  if  the  direc- 
tion of  the  axis  is  to  be  changed,  an 
angular  acceleration,  A  C,  around  an 
axis  perpendicular  to  the  original  axis 
and  intersecting  it  must  be  applied. 
The  resulting  angular  velocity  will  be 
the  sum  of  A  B  and  A  C  or  A  D  ;  and 
the  new  axis  obviously  lies  in  the  same 
plane  as  the  original  axis  and  the  axis 
of  the  acceleration.  Unless  there  is,  c"  FIG*] 
then,  an  angular,  acceleration,  neither 
the  angular  speed  nor  the  direction  of  the  axis  of  rotation 


30  THEORY   OF  PHYSICS  [CH.  I 

changes.  So  that  if  it  is  desired  to  have  a  body  move  in 
such  a  way  as  to  keep  some  plane  in  it  always  in  the  same 
direction,  all  that  is  necessary  is  to  give  it  a  rotation 
around  an  axis  perpendicular  to  this  plane.  For,  then, 
unless  an  angular  acceleration  is  impressed,  this  axis  will 
keep  its  direction  fixed.  Thus  a  quoit  thrown  at  random 
will  turn  over  and  over ;  but  if  it  be  given  a  twist  as  it 
is  thrown,  it  will  move  with  the  axis  remaining  parallel 
to  itself.  There  are  many  other  illustrations,  such  as  the 
twist  given  projectiles  from  guns,  the  twist  given  knives 
when  tossed  by  a  juggler,  etc. 

25.  3.  Harmonic  Motion.  There  is  also  harmonic  motion 
of  rotation,  as  is  illustrated  by  the  balance-wheel  of  a 
watch,  a  magnetic  compass-needle  when  slightly  displaced, 
a  wire  one  of  whose  ends  is  clamped  and  whose  other  end 
is  twisted  and  then  allowed  to  vibrate.  For  rotation,  it 
may  be  defined  as  such  motion  that  the  angular  accelera- 
tion at  any  instant  is  always  towards  the  central  position, 
and  its  numerical  value  is  a  constant  times  the  angle 
through  which  the  body  is  turned  ;  and,  further,  the  period 
of  vibration  equals  2  TT  divided  by  the  square  root  of  this 
constant.  The  "  amplitude  "  is  the  angle  which  marks  the 
extent  of  the  vibration. 

There  are,  of  course,  many  other  special  kinds  of  motion  ; 
but  these  are  perhaps  the  most  important. 


CHAPTER   II 

DYNAMICS 

DYNAMICS  is  that  branch  of  Mechanics  which  is  con- 
cerned with  the  properties  of  matter  in  motion  and  with 
the  conditions  under  which  this  motion  may  be  produced 
or  changed. 

26.  Measurement  of  Mass  by  means  of  Inertia.  That  prop- 
erty of  matter  which  is  most  familiar  to  us,  perhaps,  is  its 
inertia.  As  already  explained,  this  is  a  name  given  to  that 
property  which  becomes  evident  to  our  muscular  sense 
when  we  change  the  motion  of  matter  in  any  way.  If 
we  push  a  barrel  along  a  floor,  or  if  we  put  a  ball  in 
motion,  we  are  conscious  of  a  definite  sensation,  which 
is  said  to  be  due  to  the  inertia  of  the  matter.  We  can 
estimate  in  a  rough  way  the  amounts  of  matter  in  differ- 
ent bodies  by  the  sensations  produced  owing  to  their  in- 
ertia ;  and  we  can  easily  tell  whether  a  box  is  empty  or  full 
by  trying  to  push  it.  We  will,  then,  define  two  bodies  as 
having  the  same  mass  if  they  have  the  same  inertia.  (We 
might  perfectly  well  define  equal  masses  as  corresponding 
to  equal  "  weights ; "  and  we  have  no  right,  a  priori,  to 
regard  these  two  definitions  as  being  identical,  although 
they  may  be  proved  to  be  so  by  experiment.)  So,  if  OUT 
senses  were  delicate  enough,  we  might  perfectly  identify 
two  equal  masses.  But  since  they  are  not  so  delicate, 
some  instrument  may  be  devised  which  is  much  more 
sensitive  and  which  can  produce  the  same  effect  as  our 
muscles.  Such  an  instrument  is  a  spiral  spring  when 
compressed;  it  can  produce  change  of  motion,  and  can 


32  THEORY  OF  PHYSICS  [CH.  II 

be  tested  most  carefully.  By  means  of  it,  it  is  theoreti- 
cally possible  to  determine  whether  two  bodies  have  the 
same  inertia,  that  is,  the  same  mass  on  the  above  defini- 
tion of  equal  masses.  The  experiment  might  be  performed 
thus :  compress  the  spiral  spring  to  a  certain  definite  point ; 
place  one  of  the  bodies  on  one  end  of  the  spring,  keeping 
the  other  end  firmly  clamped ;  allow  the  spring  to  expand, 
and  accurately  measure  the  velocity  given  the  body  after 
it  leaves  the  spring.  (It  is  necessary  to  have  the  spring 
so  arranged  that  the  body  is  projected  along  a  perfectly 
smooth  horizontal  table,  so  as  to  do  away  with  all  effects 
of  friction  and  "weight.")  Then  compress  the  spring  to 
identically  the  same  point  as  before,  place  the  second  body 
upon  it,  and  measure  the  velocity  given  it  when  it  is  pro- 
jected. In  general,  these  two  velocities  would  be  different ; 
but  it  is  theoretically  possible  to  so  change  one  of  the 
bodies  by  chipping  off  small  pieces  that  the  velocities  given 
the  two  are  the  same,  within  the  range  of  the  inherent 
errors  of  the  experiment.  These  two  bodies  would  then 
have  equal  masses  as  closely  as  they  could  be  measured. 
Combining  these  two,  it  would  be  possible  to  make  a  third 
body  have  the  same  mass  as  their  combined  mass,  that  is, 
twice  the  original  mass ;  and  so  the  process  might  be  con- 
tinued until  a  series  of  bodies  was  obtained  whose  masses 
were  1,  2,  3,  4,  5,  etc.  times  that  of  the  original  body.  By 
taking  the  original  mass  as  small  as  is  wished,  it  is  per- 
fectly possible  thus  to  make  standards  of  mass  varying  by 
small  steps  from  very  small  masses  to  very  large ;  and  the 
mass  of  any  other  body  can  be  determined  by  finding  what 
combination  of  the  standard  masses  has  the  same  inertia 
as  the  body,  using  a  delicate  spiral  spring  for  the  com- 
parison. The  scientific  standard  of  mass  is  the  gram,  as 
already  defined;  and  as  it  can  be  compared  with  this 
arbitrary  set  of  masses,  their  values  in  terms  of  the  gram 
may  be  easily  deduced ;  and  so  all  measurements  of  masses 
may  be  given  in  grams. 


27]  DYNAMICS  33 

MATTER  IN  TRANSLATION 

27.  Since  in  translation  all  the  points  of  the  body  have 
the  same  motion,  the  entire  matter  may  be  considered  as 
concentrated  at  one  point;  and  this  point  thus  endowed 
with  inertia  may  be  called  a  "  particle." 

There  is  one  general  law  in  regard  to  the  translation 
of  matter,  which  so  far  as  tested  by  experiment  seems  to 
be  absolutely  true  and  to  have  no  exception.  It  is  this : 
consider  a  number  of  bodies  of  all  kinds  in  motion  of 
translation ;  and  let  their  motion  be  entirely  free  from 
'all  external  influences ;  they  will  impinge  on  each  other 
or  will  affect  each  other  in  some  way,  so  that  their  ve- 
locities will  be  constantly  changing ;  let  their  masses  be 
mi,  m2,  m3,  etc.,  and  let  their  linear  velocities  at  any  in- 
stant be  MI,  uz,  u3,  etc.  ;  form  the  geometrical  sum  m^  ui  + 
m2  u2  +  ms  us  +  etc. ;  then  the  law  is  that  this  sum  is  the 
same  for  all  other  instants,  no  matter  how  the  individual 
velocities  change.  That  is,  the  geometrical  sum 

ml  Ui  +  w2  u2  +  m3u3  +  etc.  =  a  constant      .     (1) 

The  geometrical  sum  must  be  taken,  because  the  velocities 
may  be  in  different  directions,  and  so  cannot  be  added 
algebraically.  The  law  may  also  be  stated  in  this  way: 
let  wlt  w2,  ws,  etc.  be  the  components  of  the  linear  veloci- 
ties in  any  fixed  direction ;  then  the  algebraic  sum 

mi  wl  +  m2  Wz  +  m<}Ws  +  etc. 

is  the  same  at  all  times,  if  there  are  no  external  influences. 

This  law  cannot  be  verified  directly  in  the  most  general 
case,  but  certain  special  cases  of  it  may  be ;  and  the  law 
is  regarded  as  true  in  general,  because  every  deduction 
from  it  that  admits  of  experimental  verification  has  been 
found  to  be  true. 

Owing  to  the  importance  of  the  product  m  u,  mass  times 
linear  velocity,  it  has  been  given  a  name,  "  linear  momen- 

2 


34 


THEORY  OF  PHYSICS 


[CH.  II 


turn;"  and  this  general  law  of  nature  is  called  the  "Prin- 
ciple of  the  Conservation  of  Linear  Momentum." 

Special  Cases  of  Translation.  28.  1.  Let  there  be  only 
one  body.  The  law  states  that,  if  there  are  no  external 
influences, 

m  u  =  a  constant (2) 

But  the  mass  of  a  body  cannot  change;  that  is.  m  is  a  con- 
stant. Hence  the  velocity  of  translation  must  remain 
constant.  It  follows,  then,  that  a  body  which  has  at 
any  instant  a  certain  speed  in  a  certain  direction  will 
continue  to  move  with  the  same  speed  in  the  same  direc- 
tion, if  it  is  free  from  all  external  influence.  This  seems 
to  be  perfectly  in  accord  with  experiments.  A  special 
case  of  this  is  when  the  body  is  observed  to  be  at  rest  at 
any  instant.  Since  the  velocity  is  zero  at  that  instant,  it 
must  be  zero  for  all  other  instants ;  or  a  body  once  at  rest 
will  always  remain  at  rest,  if  left  to  itself. 

29.  2.  Let  there  be  two  bodies.  Then,  if  there  is  no 
external  action,  that  is,  if  these  bodies  are  influenced  only 
by  each  other, 

+  mz  uz  =  a  constant    ....     (3) 

A  special  case  of  this  deduction  may 
be  verified  in  this  way  :  — 

Suspend  two  spheres  by  cords  so 
that  they  may  each  swing  freely  in 
vertical  circles.  Make  the  radii  of 
these  circles  the  same  length,  and 
hang  the  spheres  side  by  side.  Now, 
if  they  are  drawn  one  side  and  then 
allowed  to  impinge,  each  will  have 
a  certain  velocity  before  impact  and 
a  definite  velocity  immediately  after. 
These  velocities  are  in  the  same 
straight  line,  if  the  bodies  are 
spheres  (and  also  in  other  cases  if  the  bodies  are  of  suit- 


Em.  15. 


30]  DYNAMICS  35 

able  shapes) ;  and.  as  will  be  shown  later  on,  they  can  be 
easily  measured.  The  masses  can  also  be  measured ;  and 
so  it  can  be  proved  that  the  sum  mi  ui  +  m2  u2  is  the  same 
before  and  after  impact,  although  the  individual  velocities 
change. 

Another  experiment  which  also  verifies  the  law  is  this : 
if  a  man  standing  on  a  board  which  rests  on  a  smooth 
table  jumps  off,  the  board  goes  in  the  opposite  direction ; 
and  if  mi  is  the  mass  of  the  man,  and  ui  the  velocity  with 
which  he  jumps,  while  m2  and  u2  are  the  mass  and  velocity 
of  the  board,  then  it  is  found  that 

mi  Ui  =  —  m2  u2. 

This  is  in  agreement  with  the  law;  because,  before  the 
man  jumps,  he  and  the  board  are  at  rest,  and  hence  the 
two  velocities  are  at  that  instant  zero.  So  the  sum 

mi  Ui  +  m,2  u2 

must  always  be  zero,  if  the  only  action  is  one  between  the 
board  and  the  man.  Consequently, 

mi  Ui  +  m2  u2  —  0, 
or  mi  Ui  =  —  m2  u2, 

which  means  that  the  velocity  of  the  board  is  opposite  to 

that  of  the  man,  and  that  its  numerical  value  is  —  HI. 

m2 

An  exactly  similar  illustration  is  afforded  when  a  bullet 
is  fired  from  a  gun,  or  when  a  body  falls  towards  the  earth. 

The  illustrations  of  this  law  might  be  multiplied  indefi- 
nitely ;  but  it  is  sufficient  to  say  that  no  exception  to  it  is 
known,  and  hence  it  is  believed  to  be  one  of  the  great  laws 
of  nature. 

30.  In  terms  of  momentum,  this  law  may  be  thus 
stated :  the  sum  of  the  linear  momenta  of  any  number  of 
bodies  in  any  fixed  direction  is  a  constant,  provided  that 
no  influence  external  to  the  system  is  acting  on  it. 


36  THEORY   OF  PHYSICS  [CH.  II 

In  general,  however,  there  is  some  external  action  ;  and 
then  the  momentum  may  change.  Thus,  owing  to  the 
presence  of  the  earth,  there  is  always  a  change  in  the  mo- 
mentum of  any  piece  of  matter  which  is  moving  near  it ;  a 
piece  of  iron  near  a  magnet  has  its  momentum  changed,  if 
it  is  free  to  move,  owing  to  the  presence  of  the  magnet ;  a 
feather  moving  through  the  air  has  its  momentum  changed, 
owing  to  the  presence  of  the  air.  The  rate  of  change  of 
the  linear  momentum  of  any  body  with  respect  to  the 
time  is  called  the  "  force  acting  on  it."  Linear  momentum 
is  m  u;  but  the  mass  cannot  change,  and  the  rate  of  change 
of  the  linear  velocity  is  the  linear  acceleration  ;  conse- 
quently, force  is  the  product  of  mass  and  linear  accelera- 
tion. Therefore  the  mathematical  expression  for  force  is 

f=ma (4) 

A  unit  force  is  called  a  "  dyne."  Thus,  if  m  =  1,  and  a  —  1, 
/  —  1  ;  that  is,  if  the  mass  is  1  gram,  and  if  in  1  second 
the  velocity  is  changed  by  an  amount  1  centimetre  per 
second,  the  force  is  one  dyne.  As  an  illustration,  it  is 
found  by  experiment  that,  when  a  body  falls  towards  the 
earth,  the  acceleration  is  nearly  980,  i.  e.  that  in  each  sec- 
ond the  velocity  is  increased  by  nearly  980  centimetres 
per  second ;  hence,  if  a  mass  10  grams  falls  with  this 
acceleration,  the  force  is  9800  dynes. 

If  the  rate  of  change  of  momentum  is  uniform,  the  force 
is  the  change  of  linear  momentum  in  one  second.  Some- 
times, however,  it  is  impossible  to  measure  the  rate  of 
change  of  momentum,  e.  g.  when  a  body  is  struck  a  sud- 
den blow ;  but  it  is  always  possible  to  measure  the  total 
change ;  and  this  total  change  of  the  linear  momentum  is 
called  the  "  linear  impulse." 

Both  forces  and  impulses  have  definite  directions  and 
definite  numerical  values  ;  and  so  they  can  be  represented 
by  straight  lines.  Thus  forces  can  be  added,  subtracted, 
and  resolved  into  components.  It  may  happen  that  a  body 


32] 


DYNAMICS 


37 


has  two  forces,  as  a  piece  of  iron  near  the  earth  and  near  a 

magnet.     Let  one  force  be  represented  by  A  B,  the  other 

by  A  C ;  the  actual  force  will  be  their 

sum,  Z#.     And  if  ~AB  and  ^TC  are 

at  right  angles  to  each  other,  A  B  is 

the  component  of  A  D  in  the  direction 

of  A  to  B. 

31.  There  is  an  acceleration  produced 
in  the  motion  of  a  body  when  it  is 
allowed  to  fall  towards  the  earth,  that 
is,  there  is  a  force ;  and  so  sometimes 
it  is  said  that  the  earth  "  exerts  a  force 

on  "  the  body,  or  that  the  body  is  acted  FlG  16 

on  by  the  force  of  the  earth.  Simi- 
larly, a  magnet  is  said  to  exert  a  force  on  a  piece  of  iron. 
So  we  speak  of  the  "  force  of  the  earth's  gravity,"  the 
••'  magnetic  force,"  etc.  But  these  expressions  simply  mean 
that  under  definite  conditions  the  linear  momentum  of  a 
body  is  changed  by  a  definite  amount  per  second.  When 
a  body  is  "  acted  upon  by  "  two  forces,  each  force  produces 
its  own  effect ;  and  the  actual  force  is  the  geometrical  sum 
of  the  two,  as  proved  above.  Similarly  for  three  or  more 
forces. 

Another  illustration  is  afforded  by  a  body  which  is 
suspended  from  a  vertical  spiral  spring.  The  spring  is 
stretched  by  the  body ;  but,  since  everything  is  at  rest,  we 
say  that  the  force  of  the  earth  down  is  balanced  by  the 
force  of  the  spring  up  ;  meaning  that  if  the  spring  were 
not  there,  there  would  be  a  definite  force,  m  a,  downward ; 
and  if  the  earth  were  removed,  there  would  be  a  force 
upward,  —  ma;  and  so  the  total  force  is  zero.  A  spiral 
spring  can  thus  be  used  to  compare  and  measure  any 
forces.  Similarly,  if  a  body  is  at  rest  on  a  table,  the  force 
down  is  balanced  by  the  resistance  of  the  table. 

32.  The  laws  of  motion  for  one  or  two  bodies,  as  above 
explained,  may  now  be  expressed  in  terms  of  forces. 


38  THEORY  OF  PHYSICS  [CH.  II 

1.  The  linear  momentum  of  a  body  will  not  change,  if 
there  is  no  force  "  acting  on  "  it. 

2.  If  there  is  a  force,  it  is  measured  by  the  rate  of 
change   of   linear  momentum.     Further,  each    force   pro- 
duces its  effect  independently  of  the  others. 

3.  If  there  are  two  bodies,  whose  linear  momenta  are 
changed  owing  to  the  presence  of  each  other,  the  change 
in  the  momentum  of  the  one  must  be  equal  but  opposite  to 
the  change  in  the  momentum  of  the  other,  because  the 
sum  of  the  two  momenta  does  not  change,  and  so  the  total 
change  must  be  zero.     Hence  it  niay  be  said  that  the  force 
which  the  first  exerts  on  the  second  is  equal  and  opposite 
to  the  force  which  the  second  exerts  on  the  first ;  or,  in 
other  words,  "  action  and  reaction  are  equal  and  opposite." 
These  statements  are  called  "  Newton's  Laws  of  Motion." 

Since  force  is  mass  times  linear  acceleration,  there  are 
two  types  of  forces  corresponding  to  the  two  types  of 
linear  acceleration  :  in  one  the  speed  is  changed,  not  the 
direction ;  in  the  other  the  direction  is  changed,  not  the 
speed. 

Special  Cases.  33.  1.  Any  body  near  the  earth  is  "  acted 
upon  by"  a  force,  mgt  where  m  is  the  mass  and  g  is  a 
quantity  which  is  constant  for  any  one  place  and  nearly 
equals  980. 

Consider  the  motion  of  two  bodies  whose  masses  are  m^ 
and  rtiv  and  which  are  connected  by  a  cord ;  the  first  of 

these  hangs  verti- 
cally, the  second 
rests  on  a  smooth 
table,  the  cord  pass- 
ing over  a  pulley,  as 
shown.  There  are 
two  forces  "  acting 

•PIG.  17.  on  "  the  body  whose 

mass  is  mi ;   one  is 
due  to  the  earth,  the  other  to  the  cord.     Call  the  force  due 


34] 


DYNAMICS 


39 


to  the  cord  T  ';  it  evidently  opposes  the  force  m1g  due  to 
the  earth.     Hence  the  actual  force  is  mi  g  —  T.     That  is, 

mi  a  =  m1  g  —  T,      .....     (5) 

where  a  is  the  actual  acceleration. 

The  forces  "  acting  on  "  the  body  whose  mass  is  m2  are  the 
force  due  to  the  earth,  the  resistance  of  the  smooth  table 
to  this  force,  and  the  tension  or  pull  of  the  cord.  The 
first  two  neutralize  each  other  ;  the  third  equals  T,  because 
"action  and  reaction  are  equal  and  opposite."  Further, 
m2  must  have  the  same  acceleration  as  mi,  since  they  are 
connected  by  the  cord.  Hence 

m?  a  =  T  .     .....  -  .     (6) 

Combining  (5)  and  (6),  it  follows  that 


-9 


(7) 


That  is,  the  acceleration  is  a  constant,  and  is  in  the  direc- 
tion of  motion.  Hence  Formulae  (1),  (2),  (3)  of  Chapter  I. 
can  be  applied  to  find  the  resulting  speed 
and  distance  travelled. 

34.  2.  Let  two  bodies  whose  masses  are 
mi  and  mz  be  connected  by  a  cord  which 
passes  over  a  pulley ;  and  let  the  bodies 
hang  vertically.  If  mx  is  greater  than  m2, 
the  first  body  will  move  down  ;  the  second 
up. 

The  total  force  "  acting  down  on  "  mi  is 
ml  g  —  T.  Hence 

mi  a  =  mi  g  -  T    .     .     .     (8) 
The  total  force  acting  up  on  mz  is  T  —  m2  g. 

Hence 

mz  a  =  T  -  m2  g    .     .     .     (9) 

since  the  acceleration  of   m2  upward   equals   that   of   ma 
downward.     Combining  (8)  and  (9), 


FIG.  18. 


40  THEORY  OF  PHYSICS  [CH.  II 

g  =  m*-%.  (10) 

mi  +  m* 

And  the  resulting  speed  and  distance  travelled  may  be  at 
once  calculated. 

35.    3.  Let  two  bodies,  whose  masses  are  ml  and  m2,  be 
connected  by  a  cord  and  be  made  to  revolve  around  each 
other  by  means  of  a  "  whirling-table."     This  is  a  piece  of 
apparatus  in  which  the  two  bodies 
are  carried  by  a  horizontal  metal 
rod  which  can  be  made  by  suita- 
ble wheels  and  gearing  to  revolve 
rapidly  around  a  vertical  axis.    The 
two  bodies  have  holes  through  their 
centres,    through    which    the    rod 
FIG.  19.  passes  ;  and  so,  when  the  rod  re- 

volves,   the    bodies    turn    around 

each  other,  the  radii  of  their  paths  depending  upon  where 
the  bodies  are  placed  on  the  rod.  In  general  the  bodies 
will  not  continue  their  revolution,  but  will  slip  to  one  end 
or  the  other  of  the  apparatus.  It  is  always  possible,  how- 
ever, to  find  some  position  in  which  they  will  not  slip. 
Let  the  axis  of  revolution  be  at  the  point  0  ;  and  call  the 
distances  from  0  to  the  centres  of  the  two  bodies  r\  and  r2, 
respectively. 

Since  the  body  whose  mass  is  ml  is  moving  in  a  circle 
of  radius  i\  with  a  certain  angular  speed,  &>,  its  acceleration 
is  n  ft>2  towards  the  centre  0.  Hence  the  force  is  ml  TI  o>2. 
Similarly,  the  force  "  acting  on  "  w2  is  7??2  r*  &>2,  since  both 
bodies  are  moving  with  the  same  angular  speed.  But  action 
equals  reaction ; 

hence  mi  r\  a>2  =  ra2  rz  &>2, 

or  mlrl  =  m<ir2 (11) 

This  may  be  easily  verified  by  experiment. 


36] 


DYNAMICS 


41 


m, 


FIG.  20. 


36.  4.  As  explained  above,  since  force  is  mass  times  accel- 
eration, forces  may  be  added  and  may  be  also  resolved  into 
components.     Thus,  consider  two  bodies  whose  masses  are 
w?i   and   m2  connected 
by  a  cord  which  passes 
over  two  fixed  pulleys. 
In  between  the  pulleys 
let  a  body  whose  mass 
is  m3  be  so  suspended 
that  it  is  free  to  slide 
along  the  cord.     There 
is  a  force,  mi  g,  "  acting 
on  "  m3  in  the  direction 
of  the  cord  which  passes 
over   the   first   pulley, 

another  force,  m2g,  in  the  direction  of  the  second  pulley, 
and  also  a  force,  mzg,  vertically  downward.  Let  these 
three  forces  be  represented  by  the  lines  0  AI,  0  A2,  0  A3. 

OAi  and  0  A2  combine  to  pro- 
duce the  force  0  B.  If  this  is 
equal  and  opposite  to  0  A3,  the 
three  forces  will  neutralize  one 
another ;  and  there  will  be  no  ac- 
celeration. Consequently,  if  the 
body  whose  mass  is  ms  is  allowed 
to  come  to  rest  on  the  cord,  the 
directions  of  the  two  sections  of 
the  cord  must  be  such  that  the 
geometrical  sum  of  0  A^  and  0  A2 
is  exactly  equal  and  opposite  to 
0  A 3.  This  may  be  verified  by 
experiment. 

Another  way  of  regarding  this  problem  is  as  follows  : 
Let  the  directions  of  the  forces  be  as  shown,  when  the 
mass  ms  comes  to  rest.  The  force  0  A\  can  be  resolved 
into  two  components,  one  in  the  direction  of  the  line  OA3 


42 


THEORY   OF   PHYSICS 


[CH.  II 


reversed,  and  the  other  at  right  angles  to  it ;  i.  e.  the  two 
components  are  0  PI  and  1\  AI.     Similarly  the  force  0  A2 

can  be  resolved  into  two  compo- 
nents, 0  P2  and  P2A2.  Then, 
since  the  point  0  does  not  move, 


0  A8  =  -  (0  1\  +  0  P2). 

But  these  conditions  are  satisfied 
only  if  the  parallelogram  formed 
on  the  two  lines  0  AI  and  0  A2 
is  such  that  the  diagonal  from 
0  is  equal  and  opposite  to  OA3. 

37.  5.  Consider  the  motion  of 
two  bodies  whose  masses  are  m1 
and  m2)  and  which  are  connected 

by  a  cord  passing  over  a  pulley,  so  that  one  body  hangs 

vertically  arid  the  other  rests  on 

a  smooth  plane  inclined  to  the 

horizon  at  an  angle  9.     The  force 

of  the  earth  on  m2  acts  vertically 

down  ;  its  component  parallel  to 

the  plane  is  m2g  sin  0.     Hence, 

if  T  is  the  tension  of  the  string, 

the  actual  force  on  m2  down  the 

plane  is 


FIG.  23. 


sin  9  -  r, 
and  the  force  on  m\  is 

m\  a  —  T  —  mi  g. 

m2  sin  9  —  m\ 


Hence 


m\ 


9 


(12) 


38.  Centre  of  Inertia.  Let  the  masses  of  a  system  of 
particles  be  m1}  m2,  ms,  etc. ;  and  let  their  distances  at  any 
instant  from  a  fixed  plane  be  xi,  x2)  zs,  etc.  A  special  case 


38]  DYNAMICS  43 

of  this  is,  of  course,  a  single  body  like  a  stone  or  a  drop  of 
water,  where  MI,  m2,  etc.,  are  the  masses  of  the  small  por- 
tions into  which  the  body  may  be  considered  divided. 
Form  the  algebraic  sum 

mi  Xi  +  w2  xa  +  ms  xs  +  etc. 

and  divide  this  by  mi  +  m2  +  m8  +  etc.,  i.  e.  by  the  mass 
of  the  system.  This  quotient  gives  obviously  the  average 
distance  of  the  entire  system  from  the  fixed  plane.  Call 
this  distance  x.  Then 

(mi  +  m2  +  m3  +  etc.)  x  =  mv^  +  m2x2  -f  m3xs  +  etc.  (13) 


Imagine  these  masses  all  in  motion  away  from  the  fixed 
plane  ;  each  will  have  a  definite  velocity  of  its  own  which 
is  the  rate  of  change  of  its  distance  from  the  plane.  Writ- 
ing u  for  the  rate  of  change  of  x,  u\  for  the  rate  of  change 
of  Xi,  etc.  (13),  gives  at  once 

(mi  +  m2  +  m3  +  etc.)  u  =  mlul  +  m2u2  +  mzuB  +  etc.  (14) 
If  there  are  no  external  forces,  equation  (1)  gives 
mi  HI  +  w2  uz  +  m8  u8  +  etc.  =  a  constant. 


Hence  u  is  a  constant  ;  that  is,  the  average  distance  of 
the  matter  from  the  fixed  plane  increases  at  a  constant  rate. 
Of  course,  ui,  u2,  etc.,  may  change,  owing  to  the  action  of 
the  different  particles  on  each  other  ;  but  these  changes 
are  such  that  %  does  not  change. 

If  there  are  external  forces,  there  will  be  changes  in  the 
velocities  Ui,  u2,  etc.,  owing  to  the  external  action  in  addition 
to  those  due  to  the  mutual  action  of  the  bodies.  Conse- 
quently u  will  also  change.  Let  a  be  the  rate  of  change  of  u, 
ai  that  of  Ui,  a2  that  of  u2,  etc.  Hence,  from  (14)  it  fol- 
lows that 

(/H!  +  mz  +  ms  +  etc.)  a  =  m-^  a\  +  m2a2  +  ms  a8  +  etc. 

=  Xi  +  Xz  +ZS  +  etc., 


44  THEORY   OF  PHYSICS  [CH.  II 

where  Xi  is  the  entire  force  of  mlt  in  the  direction  away 
from  the  fixed  plane,  including  both  external  and  internal 
forces  ;  X^  applies  similarly  to  m2,  etc.  But  the  sum  of  all 
the  internal  forces  in  any  fixed  direction  equals  zero,  be- 
cause "  action  and  reaction  are  equal  and  opposite."  Hence 
Xi  +  X%  +  Xz  4-  etc.  equals  simply  the  sum  of  the  exter- 
nal forces  in  the  direction  away  from  the  fixed  plane  ;  and 
it  may  be  written  X.  Consequently, 

(mi  +  m2  +  mz  +  etc.)  a  =  X,     .     .     .     (15) 

which  states  that  the  acceleration  away  from  the  plane  of 
the  average  distance  of  the  system  from  the  plane  is  equal 
to  the  total  external  force  in  that  direction  divided  by  the 
total  mass. 

39.  We  might  in  a  similar  way  consider  the  average  dis- 
tance of  the  system  (or  the  single  body)  from  any  other 
plane.  In  particular,  let  us  take  three  fixed  planes  at  right 
angles  to  each  other,  e.  g.  the  three  planes  meeting  in  the 
corner  of  a  room,  and  find  the  average  distances  of  the 
systems  from  them,  and  the  changes  in  these  distances 
We  may  write 


(mi  4-  ?7?  2  4-  w»  +  etc.)  x  =  mi  Xi  +  m2  a?2  -\-  m^xz-\-  etc.  "j 
(mi  4-  m2  +  ms  4-  etc.)  y  —  mi  y^  4-  mz  ?/2  +  mB  y*  -f  etc.  >  (16) 
(mi  +  m2  +  w3  4-  etc.)  z  =  mi  Zi  +  m2  z2  +  m8  zs  +  etc.  J 

where  the  x,  y,  z's  refer  to  the  distances  from  the  three 
planes.  But  since  x,  y,  z,  have  definite  numerical  values 
at  any  instant,  they  determine  the  position  of  a  point 
whose  distance  from  one  plane  is  x,  i.  e.  the  average  dis- 
tance of  the  entire  system  from  that  plane,  and  similarly 
for  the  other  planes.  Consequently  this  point  represents 
the  average  distance  of  the  system  from  the  three  fixed 
planes  ;  and  it  is  called  the  "  centre  of  inertia  "  of  the  sys- 
tem. It  is  not  a  point  of  the  body,  but  a  geometrical  point 
in  space. 


39]  DYNAMICS  45 

Thus  the  centre  of  inertia  of  a  uniform  rod  is  its  middle 
point.  For,  consider  the  rod  made  up  of  equal  separate 
masses,  and  let  mi  and  m2  be 
two  which  are  at  the  ends. 
Take  as  the  plane  of  refer- 
ence one  perpendicular  to  the 
rod,  and  let  Xi  and  0%  be  the 
distances  of  mi  and  m2  from 
the  plane. 


m, 


FlG  24 


(ml  +  raa)  x  =  mi  Xi  +  m2  x2. 


_  m    _,      . 

But  mi  =  mz  ;  hence  x  =  -^    —  ,  i.  e.  the  centre  of  inertia 

'Z 

of  these  two  masses  is  half-way  between  them.  Similarly 
for  the  other  masses  which  make  up  the  rod,  always  com- 
bining those  which  are  equidistant  from  the  two  ends. 
The  centre  of  inertia  of  a  uniform  sphere  (or  spherical 
shell)  is  also  its  centre  of  figure.  Other  illustrations  will 
be  given  later. 

The  velocity  of  the  centre  of  inertia  away  from  the  three 
planes  is  given  by  the  three  equations 

(MI  +  m2  +  w8  +  etc.)  u  =  mv  KI  +  m2  u2  +  m8  ?/8+  etc.  "j 
(mi  +  w2  +  mz  +  etc.)  v  =  mi  Vi  +  m2  v2  +  m3  vs  +  etc.  >  (17) 
(mi  +  m2  +  rtiz  +  etc.)  w  =  miWi  +  m2w2  +  msiv8  +  etc.  ) 

And  the  actual  resultant  velocity  is  found  by  adding  u,  v, 
w,  geometrically,  so  as  to  give  the  diagonal  of  a  rectangular 
parallelepiped  of  which  they  are  the  edges. 

If  there  are  no  external  forces,  u,  v,  w  are  constant  ;  so, 
also,  is  their  geometrical  sum.  Consequently,  the  centre  of 
inertia  will  move  in  a  straight  line  with  a  constant  speed, 
or  will  remain  at  rest.  As  an  illustration  of  this  fact,  it  is 
evident  that  if  any  number  of  bodies  are  at  rest,  and  are  then 
set  in  motion  by  their  mutual  actions,  they  will  so  move  that 
their  centre  of  inertia  always  remains  the  same  as  it  was 
before  the  motion.  Thus,  when  a  bullet  is  fired  from  a 


46  THEORY   OF   PHYSICS  [CH.  II 

rifle  so  suspended  as  to  be  free  to  move,  the  resulting  mo- 
tion will  be  such  as  to  leave  the  centre  of  inertia  of  the 
rifle  and  the  bullet  unchanged.  If  a  boiler  explodes,  the 
parts  will  so  move  that  the  centre  of  inertia  does  not 
change.  If  the  solar  system,  i.  e.  the  sun  and  planets  with 
their  satellites,  is  free  from  external  action,  its  centre  of 
.inertia  must  be  moving  through  space  with  a  constant 
velocity,  if  it  is  in  motion  at  all. 
If  there  are  external  forces, 

(mi  +  mz  +  m8  -f  etc.)  a  =  X  \ 

(mi  +  ms  +  ms  +  etc.)  b  =  Y  I  .     .     .     (18) 

(mi  +  m2  +  ms  +  etc.)  c  =  Z  J 

where  a,  b,  ~c  are  the  accelerations  of  the  centre  of  inertia 
in  the  three  directions ;  and  X,  Y,  Z  are  the  sums  of  the 
external  forces  in  those  directions,  because  all  the  internal 
forces  are  balanced.  The  actual  acceleration  of  the  centre 
of  inertia  is  the  geometrical  sum  of  a,  b,  c. 

X  is  the  sum  of  the  components  in  a  certain  direction  of 
all  the  external  forces  "  acting  on  "  each  one  of  the  parti- 
cles ;  similarly  Y  and  Z  refer  to  the  components  in  the 
other  two  directions.  If  these  components  were  to  act  at 
a  single  point,  they  would  have  a  resultant  effect  equal 
to  their  geometrical  sum.  Consequently  adding  geometri- 
cally the  three  equations,  and  writing  M  for  the  entire 
mass,  A  for  the  actual  acceleration  of  the  centre  of  inertia, 
and  R  for  the  geometrical  sum  of  X,  Y,  Z, 

MA  =  R (19) 

This  equation  states  that,  when  there  are  external  forces, 
the  motion  of  the  centre  of  inertia  of  a  system  of  bodies  (or 
a  single  body)  is  exactly  the  same  as  would  be  that  of  a 
particle  of  mass,  M,  "  acted  upon  by  "  a  force,  R,  where  M 
is  the  total  mass  of  the  system,  and  R  is  the  geometrical 
sum  of  all  the  external  forces.  Or,  so  far  as  translation  is 
concerned,  the  motion  of  any  body  is  exactly  the  same  as 


40]  DYNAMICS  47 

would  be  that  of  a  particle  whose  mass  equals  the  entire 
mass,  if  "  acted  upon  "  by  all  the  external  forces  directly. 

Thus,  consider  a  uniform  stick  thrown  at  random  along 
a  horizontal  table.  The  centre  of  inertia  of  the  stick  is  its 
middle  point;  and,  as  the  stick  moves,  this  point  must 
follow  a  straight  line,  however  the  stick  itself  may  revolve  ; 
because,  if  a  body  whose  mass  were  equal  to  that  of  the 
stick  were  acted  upon  by  a  force  equal  to  that  used  on  the 
stick,  it  would  move  in  a  straight  line. 

Again,  consider  a  bomb-shell  fired  in  the  air.  If  it  does 
not  burst,  its  centre  of  inertia,  i.  e.  its  centre  of  figure,  will 
describe  a  certain  path.  But  even  if  it  does  burst,  the 
fragments  will  so  move  that  their  centre  of  inertia  will  fol- 
low identically  the  same  path,  because  the  external  forces 
are  not  changed. 

40.    Special  Cases  of  Centres  of  Inertia. 

1.  Triangular  board,  ABC. 

Draw  the  three  medial  lines  Aa,Bl,Ccy  connecting  the 
vertices  with  the  middle  points  of  the  opposite  sides.  They 

meet  in  a  point  0.     Since  the 
straight  line  A  a  divides  the 
triangle  into  two  equal  halves, 
the  centre  of  inertia  must  lie 
on  it ;  for  the  triangle  may  be 
considered  built  up  of  a  great 
number  of   strips  parallel   to 
the  side  B  C,  and  as  the  centre 
of  inertia  of  each  of  these  strips 
lies   on  the  medial  line  A  a, 
the  centre  of  inertia  of  the  en- 
tire triangle  must  lie  on  it  also.     Similarly,  it  must  lie  on 
B  b  and  C  c  ;  that  is,  it  must  be  the  point  0,  their  common 
point  of  intersection. 

2.  A  uniform  rod,  mass  m3  =  25,  carrying  two  symmet- 
rical bobs  whose  masses  are  Wi  =  15,  m2  =  20 ;  the  dimen- 
sions and  distances  being  as  indicated  (Fig.  26). 


48 


THEORY   OF   PHYSICS 


[CH.  II 


The  centre  of  inertia  of  the  rod  itself  is  its  middle  point  ; 
that  is,  the  rod  acts  as  if  its  mass  were  concentrated  at 
that  point,  which  is  at  a,  distance  15  cm.  from  the  ends. 
Take  as  a  plane  from  which  to  measure  distances  one 
perpendicular  to  the  rod  at  its  left  end.  Then 

mi  =  15,  Xi  =  5  ;     ra2  =  20,  x2  '=  20  ;     m3  =  25,  x9  =  15  ; 
+  m2x2  +  msxs      75  +  400  +  375 


, 
ana 


_ 

x 


mi  +  mz  +  ms 


— 
60 


—  14.  1  /. 


The  centre  of  inertia  must,  then,  lie  at  a  distance  of  14.17 
cm.  from  the  plane  at  the  end  of  the  rod  ;  and  since  the 
bobs  are  symmetrical,  it  must  lie  in  the  axis  of  the  rod  at 
that  distance  from  the  end. 


FIG.  27. 

3.  A  rigid  framework  lying  in  a  plane ;  two  bodies, 
whose  masses  mi  =  20,  m2  =  10,  are  connected  by  massless 
wires  to  a  uniform  rod  whose  mass  ms  =  10 ;  the  dimen- 
sions being  as  shown. 

Take  as  the  two  planes  of  reference  one  perpendicular 
to  the  rod  at  its  lower  end,  the  other  through  the  rod  per- 
pendicular to  the  two  wires. 


Hence 


So 


mi  =  20, 
m2  =  10, 
ms  =  10, 


=  10; 


x,  =  20, 
xs  =  10, 


7/3   = 


x  =  Yif  =  7.5  ;     y  =  %°-  =  6.25. 


, 
41]  DYNAMICS  49 

That  is,  the  centre  of  inertia  is  a  point  at  a  distance  7.5  cm. 
from  the  plane  perpendicular  to  the  rod  at  its  lower  end ; 
and  a  distance  6.25  cm.  from  the  rod  itself  in  a  direction 
parallel  to  the  wires ;  therefore  it  is  at  the  point  0  as 
shown. 

MATTER  IN  KOTATION 

41.  There  is  another  general  law  which  is  obeyed  by 
a  system  of  bodies  in  motion,  and  which  is  not  a  neces- 
sary consequence  of  the  law  which  has  just  been  dis- 


FIG.  28. 

cussed,  and  which  states  that  m±  u\  +  mz  uz  +  mz  u3  4-  etc., 
=  a  constant  if  there  is  no  external  influence.  This  sec- 
ond law  may  be  stated  in  this  way :  Let  mb  ra2,  ms,  etc., 
be  the  masses  of  a  system  of  particles  ;  let  their  veloci- 
ties in  any  plane  (or  in  planes  parallel  to  each  other)  be 
ult  u2,  us,  etc.,  as  shown ;  let  Si,  $2,  s3,  etc.,  be  their  speeds  ; 
and  let  0  be  the  position  of  any  fixed  line  perpendicular  to 
these  parallel  planes.  Draw  perpendiculars  from  this  line 
to  the  directions  of  the  velocities,  and  let  h,  /2, 18,  etc.,  be 
their  lengths.  Then  the  general  law  is  that 

mi  si  h  +  ^2  s2 12  +  77Z8  ss  IB  +  etc.  =  a  constant      (20) 


50  THEORY  OF  PHYSICS  [CH.  II 

if  there  are  no  external  influences.  In  this  sum,  a  product 
is  called  positive,  if  looking  at  m  from  0  it  is  moving  to- 
wards the  right ;  thus,  m1  Si  ^  and  m2  s2  12  are  positive ; 
ms  ss  13  is  negative.  The  product  m  s  I  is  called  the  "  mo- 
ment of  momentum  "  of  the  mass  in  moving  with  a  speed, 
s,  in  a  line  at  a  distance,  I,  from  the  arbitrary  axis ;  and 
this  law  is  called  the  "Principle  of  the  Conservation  of 
Moment  of  Momentum." 

This  law  is  thought  to  be  true  in  all  cases,  and  thus  to 
be  one  of  the  great  laws  of  nature,  because  no  exceptions 
to  it  are  known  ;  all  observed  phenomena  are  in  accord 
with  it. 

SPECIAL  CASES  OF  ROTATION 

42.  1.  Let  there  be  a  single  particle.  Then  m  s  I  =  a 
constant,  if  there  are  no  external  influences.  (This  might 
be  considered  also  as  a  consequence  of  the  general  law  that 
"  m  u  —  a  constant,  if  there  are  no  external  influences.") 

But  m  s  I  will  still  remain  a 
constant  when  there  are  ex- 
ternal influences,  if  these  are 
of  a  definite  kind.  Thus,  let  a 
particle  whose  mass  is  m  be 
connected  to  a  pivot,  0,  by  a 
rigid  beam  of  length,  r,  whose 
FJG>  29.  mass  is  so  small  that  it  may 

be   neglected.      The   body   will 

then  move  in  a  circle  around  the  pivot.  Measuring  /  from 
this  pivot,  I  —  r,  s  =  rco.'.msl  =  mrzco,  where  w  is  the 
angular  velocity,  because  the  product  m  s  I  is  positive  for 
motion  around  a  definite  axis  in  a  definite  direction.  Now, 
it  is  an  observed  fact  that  co  will  remain  constant,  and 
hence  m  r2  o>  also,  if  all  the  forces  acting  on  m  have  direc- 
tions which  pass  through  the  pivot.  Forces  of  this  kind 
simply  alter  the  pressure  on  the  pivot. 


43]  DYNAMICS  51 

43.  If  there  is  a  force  "  acting  on  "  m,  whose  direction 
does  not  pass  through  the  pivot,  it  is  observed  that  there 
is  a  change  in  the  angular  speed ;  i.  e.  mr^co  changes. 
The  rate  of  change  of  mr2co  is  called  the  "  moment 
around  the  axis "  to  which  r  and  o>  refer,  m  r2  cannot 
change  if  r  is  constant ;  and  the  rate  of  change  of  co  is  the 
angular  acceleration.  Call  the  moment,  L ;  the  angular 
acceleration,  a ;  and  mr2,  /.  Then 

L  =  Ia (21) 

This  is  perfectly  analogous  to  the  similar  equation  in  mo- 
tion of  translation, 

F  =  m  a. 

The  moment  of  inertia,  /,  measures  inertia  of  matter  for 
rotation,  as  mass,  m,  does  for  translation.  And  it  is  seen  to 
depend  upon  not  alone  the  mass  but  also  the  distance  of 
the  particle  from  the  axis  of  rotation.  If  there  are  several 
particles  revolving  around  the  same  axis,  the  entire  mo- 
ment of  inertia  is  the  sum  of  the  individual  ones,  just  as 
the  entire  mass  is  the  sum  of  the  separate  ones.  Further, 
just  as  m  u  is  called  the  linear  momentum,  so  /  co  is  called 
the  "  angular  momentum."  That  is,  in  the  case  of  motion 
in  a  circle  around  a  fixed  axis,  the  moment  of  momentum 
is  called  the  angular  momentum. 

As  explained  in  Chapter  I.  Article  16,  angular  accelera- 
tions may  be  compounded,  if  the  axes  meet  in  a  point. 
Consequently  moments  may  also  be  added,  or  resolved  into 
components. 

A  moment  is  produced,  as  stated  above,  when  the  direc- 
tion of  the  force  does  not  pass  through  the  pivot ;  and  the 
connection  beween  moment  and  force  is  easily  found. 

Consider  a  particle  whose  mass  is  m,  revolving  in  a  circle 
of  radius,  r,  around  a  pivot,  0,  with  an  angular  velocity 
which  at  any  instant  equals  &>.  I  =  m  r2. 


52 


THEORY  OF  PHYSICS 


[CH.  II 


Hence  /&>  =  wr2o>  =  mru,  where  u  is  the  linear  velo- 
city of  m  at  that  instant.  Let  a  force,  F,  "  act "  perpendicu- 
lar to  the  radius,  F  =  m  a  ;  but  the  moment  is  the  rate  of 
change  of  m  r  u,  that  is,  L  =  m  r  .  a ;  hence 

L  =  Fr. 

Therefore  the  moment  around  an  axis  equals  the  prod- 
uct of  the  force  and  the  perpendicular  distance  from  the 


m 


FIG.  30. 


FIG.  31. 


axis  to  the  direction  of  the  force.  Similarly,  if  the  force 
F  is  applied  obliquely  to  the  radius.  The  moment,  by 
definition,  is 

L  =  ma 


r. 


The  component  of  the  force  F  perpendicular  to  the  radius 
is  Fcos  0  ;  and  this  must  equal  m  a ;  i.  e. 

F  cos  0  =  m  a. 


Hence 


L  =  FT  cos  6  =  Fl, 


(22) 


where  I  is  the  perpendicular  distance  from  the, axis  to  the 
direction  of  the  force.  It  is  not  necessary  to  take  into 
account  the  other  component  of  the  force  F,  because  it  is 
along  the  radius  and  so  passes  through  the  pivot,  simply 
producing  a  pressure  there. 


44]  DYNAMICS  53 

44.  2.  Let  there  be  any  number  of  particles  which  are 
rigidly  connected,  and  which  are  at  any  instant  turning 
around  an  axis.  An  illustration  of  this  is  an  ordinary  solid 
in  rotation. 

Let  w-i,  m2,  m3,  etc.,  be  the  masses  ;  r\>  r2,  rs,  etc.,  their 
distances  from  the  axis  ;  and  <w,  the  angular  velocity  com- 
mon to  all.  Hence  li  =  i\,  Si  =  riot',  /2  =  r2,  s2  =  r2  (o  ; 
ls  =  rs,  ss  =  r3  co  ;  etc.  ;  and  so 


4- 

^3^32  4-  etc.) 


4-  war82  +  msr32  4-  etc.  is  the  sum  of  the  moments  of 
inertia  of  m\t  M.Z,  m8,  etc.,  around  the  axis  of  rotation  ;  call 
it  the  total  moment  of  inertia  of  the  system  around  the 
axis,  and  write  it  /.  The  law  is  that  the  product 

Ico  =  a  constant      .....     (23) 

if  there  are  no  external  influences.  This  is  verified  by 
experiment. 

Thus,  if  a  fly-wheel  is  set  spinning  around  an  axle,  it 
would  continue  revolving  with  a  constant  speed  forever 
were  it  not  for  friction. 

But  if  there  are  external  influences,  7o>  can  be  altered  in 
two  ways,  —  by  the  direction  of  the  axis  changing,  and  by 
the  angular  speed  changing.  To  produce  an  angular  speed 
around  any  axis,  a  moment  around  that  axis  is  necessary, 
as  was  shown  in  Case  1.  And,  as  was  proved  in  Chapter  I. 
(Art.  24),  if  this  additional  angular  speed  is  around  the 
same  axis  as  the  existing  motion,  the  angular  speed  will  be 
changed,  but  not  the  direction  of  the  axis  ;  if  the  additional 
angular  speed  is  around  an  axis  perpendicular  to  that  of  the 
existing  motion,  then  the  direction  of  the  axis  is  changed. 
Consequently,  the  product  /  a)  will  remain  a  constant  un- 
less the  external  forces,  if  there  are  any,  produce  a  suitable 
moment. 


54 


THEORY  OF  PHYSICS 


[CH.  II 


45.  Illustrations,     a.  A  disc  rotating  around  a  fixed  axis. 
Let  the  disc  be  pivoted  at  the  point  0,  and   let   there 
be  a  force,   F,  in  the  plane  of  the  disc  applied  at  the 

point  P  (e.  g.  let  a  string  which  is 
fastened  to  a  nail  at  P  be  pulled  with 
a  force,  F)'. 

Since  the  disc  is  a  rigid  body,  the 
effect  of  the  force  F  is  the  same  when 
applied  at  P  as  if  applied  at  any  point 
in  the  line  of  the  force ;  and  its  mo- 
ment around  the  axis  0  is  Fl,  where 
I  is  the  perpendicular  distance  from 
0  to  the  line  of  action  of  F.  Then, 
if  /  is  the  moment  of  inertia  about 
the  pivot, 

FIG.  32.  Ft  =  la       .      .     .      (24) 

This  moment  in  the  case  as  shown  produces  angular  accel- 
eration in  a  direction  opposite  to  the  motion  of  the  hands 
of  a  watch.  Let  there  be  another  force  whose  moment 
around  the  axis  0  is  equal  to  the  former  moment,  but 
which  produces  acceleration  in  the  same  direction  as  the 
motion  of  the  hands  of  a  watch ;  then,  if  both  these  mo- 
ments "  act  on  "  the  body,  they  will  neutralize  each  other, 
and  the  angular  acceleration  will  be  zero ;  that  is,  the 
angular  velocity  will  remain  constant. 

46.  b.  Two  discs  which   are   so  arranged   as  to  rotate 
around  a  fixed  axis.     After  being  set  in  rotation,  let  them 
be     pushed     together. 

Then,  owing  to  friction 
between  their  surfaces,     ^^ 
they  will  gradually  as- 
sume the  same  angular 

speed.     The  two  discs  FlG  33 

form  a  system  entirely 
removed  from  external  influences,  except  the  push  which 


47] 


DYNAMICS 


brings  them  together.  This,  however,  is  a  force  along  the 
axis,  and  so  produces  no  moment.  Consequently,  accord- 
ing to  the  general  law,  the  total  angular  momentum  must 
remain  unchanged.  Let  I\  and  /2  be  the  moments  of  inertia 
of  the  two  discs  around  the  axis,  o>!  and  o>2  their  angular 
velocities  before  coming  in  contact,  o>3  the  angular  velocity 
common  to  both  after  the  speed  becomes  the  same  for 
both.  Then  it  follows  that 


+    /a  C*2  =   (/l    +    /) 


(25) 


This  is  perfectly  in  accord  with  experiment. 

The  analogy  should  be  noticed  between  this  phenomenon 
and  that  of  the  impact  of  two  bodies  which  stick  together 
after  meeting.  In  this  latter 

m-i  Ui  +  m2  Uz  —  (mi  +  m2)  u&. 

47.    c.    A  cylinder  rolling  down  an  inclined  plane. 
The  moment  in  this  case  is  due  to  the  earth's  force,  mg, 
and  the  axis  of  rotation  is  at  any  instant  the  line  of  con- 


FIG.  34. 

tact  between  the  cylinder  and  the  plane.  The  force  acts 
at  the  centre  of  inertia  of  the  cylinder  ;  and  consequently, 
calling  r  the  radius  of  the  cylinder,  the  moment  of  the 
force  about  the  axis  of  rotation  is  the  product  of  m  g  and 
the  perpendicular  distance  from  A  to  a  vertical  line  through 
the  centre  of  the  cylinder.  That  is, 


56  THEORY   OF  PHYSICS  [CH.  II 

L  =  mgr  sin  0. 

And  so  mgr  sin  0  =  la. 

It  may  be  proved  by  higher  mathematics  that  in  this  case 


TT  o     sin  0 

Hence          .  a  —  f  g  -  - 


Thus  a  is  a  constant ;  and  as  the  direction  of  the  axis  does 
not  change,  the  resulting  angular  speed  and  angle  turned 
through  may  be  calculated  from  Formulse  (10),  (11),  and 
(12),  Chapter  I.  Article  22. 

48.  d.  A  spinning  top  whose  axis  is  not  vertical.    Owing 
to  the  action  of  the  earth,  there  is  a  moment  tending  to 

make  the  top  turn  over ;  that 
is,  the  moment  is  around  an 
axis  perpendicular  to  the  axis 
of  figure.  If  the  top  is  spinning, 
the  motion  will  then  be  com- 
pounded of  the  velocity  around 
the  axis  of  figure  and  an  accel- 
eration around  an  axis  perpen- 

dicular  to  it.     Hence  the  motion 

FlG  35  is   as    described    in    Chapter   I. 

Article  23. 

49.  Centre  of  Inertia.     If  the  only  force  acting  on  a  sys- 
tem of  rigidly  connected  particles  is  applied  in  such  a 
direction  as  to  pass  through  the  centre  of  inertia,  the  sys- 
tem will  not  be  set  in  rotation  (or,  if  it  has  any  angular 
velocity,  this   will   not   be   altered).     A   special    case,   of 
course,  of  such  a  system  is  a  single  solid  body. 

This  theorem  may  be  proved  in  this  way :  Let  F  be  a 
force  whose  direction  passes  through  the  centre  of  inertia 
of  the  system ;  mi,  m2,  ms,  etc.,  the  masses  of  the  differ- 
ent particles  composing  the  rigid  system ;  x\,  xz>  xa,  etc., 


49] 


DYNAMICS 


57 


their  distances  from  a  fixed  plane  which  is  parallel  to  the 
direction  of  the  force  ;  «i,  a2,  as,  etc.,  their  linear  accelera- 
tions parallel  to  the  direction  of  the  force ;  and  0  the 


FIG.  36. 


centre  of  inertia  at  a  distance,  x,  from  the  plane.     Then, 
from  (16)  and  (18), 

+  m2  x2  -+-  msxs  +  etc. 


_ 

mi  +  m2  H-  w3  +  etc. 

-      mi  cti  -{-  m2a2  -{-  ms  «8  +  etc. 
???  i  -f  w2  +  ms  +  etc. 

The  sum  of  the  moments  of  the  forces  acting  on  each 
particle,  taken  around  any  axis  which  is  in  the  fixed  plane, 
and  which  is  perpendicular  to  the  direction  of  the  force,  is 

rm.\aJ\x\  +  m2  a2  X2  +  w3  as  xs  +  etc. 

and  this  moment  of  the  system  must  equal  the  moment 
of  the  force  around  the  same  axis,  Fx.  From  (19) 

F  =  (mi  +  m2  +  mB  +  etc.)  a. 
Hence 

(mi  +  m2  +  ms  +  etc.)  ax  =  m-\aiXi  +  ni2a2x2  +  msasx8  +  etc. 


58  THEORY  OF  PHYSICS  [CH.  II 

That  is,  substituting  for  x  its  value, 

?»!  a  a?!  +  mz  ax2  +  ms  ax3  +  etc.  = 

Wj  ttj  xl  +  ra2  #2  #2  +  ms  a3  xs  +  etc. 

And  as  this  same  equation  must  hold  for  different  values 
of  the  x 's  (because  the  plane  is  any  fixed  plane  parallel  to 
the  force),  their  coefficients  on  the  two  sides  of  the  equa- 
tion must  be  identical ;  that  is, 

a  =  al  =  a2=  a3  —  etc.,    ....     (26) 

which  means  that  all  the  portions  of  the  system  have 
the  same  linear  acceleration,  so  that  there  is  no  rotation 
produced. 

50.  If  the  force  is  applied  in  a  direction  which  does  not 
pass  through  the  centre  of  inertia,  there  will  be  both  trans- 
lation and  rotation ;  the  centre  of  inertia  will  move  as  if 
the  entire  mass  were  concentrated  there  and  "  acted  upon  " 
by  the  total  force,  and  the  body  will  itself  rotate  around 
the  centre  of  inertia  as  it  moves.     It  may  be  proved  that 
the  rotation  produced  around  axes  passing  through  the 
centre  of  inertia  as  it  is  moving  is  exactly  the  same  as  it 
would  be  if  the  centre  of  inertia  was  fixed. 

HARMONIC  MOTION 

51.  An  illustration   of  harmonic  motion  is  given  by  a 
simple  pendulum,  which  consists  of  a  particle  of  mass,  m, 
suspended  from  a  fixed   point,  P,  by  a  string  of  length,  /, 
whose  mass  is  too  small  to  be  taken  into  account.     If  the 
pendulum  is  set  in  vibrations  in  a  vertical  plane,  there  will 
be  a  moment  due  to  the  earth's  action  which  will  always 
tend  to  bring  the  pendulum  back  into  its  vertical  position. 
As  was  stated  in  Article  33,  there  is  a  force,  mg,  acting  ver- 
tically down  on  m ;  and  hence,  when  the  angular  displace- 
ment of  the  pendulum  is  0,  the  moment  around  the  axis  P 


51]  DYNAMICS  59 

is  m  g  I  sin  6.  The  moment  of  inertia  of  m  around  P  is 
m  I2.  Hence,  since  the  angular  acceleration  is  the  ratio  of 
the  moment  of  the  force  to  the  moment  of 
inertia  (Art.  43),  its  value  is 


m  /2  I 

But  if  the  angle  is  small,  sin  6  =  6.     So  the 

angular  acceleration  is  |  6  ;  and  since  it  op- 

l 

poses  displacement  from  the  vertical  position, 
the  vibrations  are  harmonic  (Art.  25).  The 
period,  then,  for  small  vibrations  is 


(27) 


It  is  possible  that  g  might  be  different  for  different  masses 
or  for  different  kinds  of  matter  ;  but  this  is  proved  by 
direct  experiment  not  to  be  true. 

Consequently  a  pendulum  of  any  kind  may  be  used  to 
measure  equal  intervals  of  time,  since  its  period  is  a  con- 
stant, depending  only  on  its  length. 

The  "second's  pendulum"  is  a  pendulum  which  "beats  " 
seconds  ;  that  is,  its  complete  period  is  2  seconds.  Hence, 
T  =  2  ;  and,  if  g  =  980, 

T*g       980 
I  =  --    =  -     =  99.3  cm. 


A  pendulum,  then,  of   this  length,   swinging   at  a  place 
where  g  =  980,  would  make  a  half  vibration  each  second. 

The  motion  of  a  simple  pendulum  may  be  regarded  also 
as  a  case  of  translation  ;  and,  if  the  vibration  is  small 
enough,  the  arc  is  practically  a  straight  line.  The  force 
which  tends  to  bring  the  pendulum  back  into  its  vertical 
position  is  mg  sin  6  ;  so  the  linear  acceleration  is  g  sin  6 


60 


THEORY  OF  PHYSICS 


[CH.  II 


or  g  0,  if  the  arc  is  small.  The  linear  displacement  of  the 
pendulum  bob  from  its  vertical  position  is  /  0.  Call  this  x. 
Then 

a  =  g  6 

x  =  1  0 


linear  acceleration 
linear  displacement 


q 
—  ~ 


And  as  the  acceleration  is  always  towards  the  central 
position,  the  motion  is  harmonic  (see  Art.  21);  and  the 
period  of  the  small  vibrations  is 


9 

The  tension  in  the  string  does  not  influence  the  motion 
in  the  least,  as  its  line  of  action  passes  through  the  pivot, 
and  so  its  moment  is  zero. 


MOTION  IN  GENERAL  OF  A  RIGID  BODY 

52.    It  has  been  proved  (Art.  36)  that,  if  a  body  is  so 
small  that  it  can  be  considered  a  particle,  its  motion  under 

the  action  of  two  forces, 
represented  by  the  lines 
0  AI  and  0  A 2,  is  given 
by  their  sum,  a  force 
represented  by  0  B. 
Further,  in  the  case  of 
a  large  body,  it  has  been 
proved  that,  so  far  as 
translation  is  concerned, 

the  motion  of  the  centre  of  inertia  is  just  what  the  motion 
of  a  particle  would  be  if  its  mass  were  that  of  the  body, 
and  if  all  the  forces  acted  on  it  directly.  The  motion  of 
rotation  of  a  large  body  is  also  known,  if  there  is  but  a 


FIG.  38. 


53] 


DYNAMICS 


61 


single  force  acting  on  it.  The  problem  now  is  to  find,  not 
alone  the  motion  of  translation,  but  also  that  of  rotation, 
when  a  large  body  is  acted  on  by  any  number  of  forces. 

This  general  problem  is  quite  difficult ;  and  so  only  a 
special  case  will  be  considered  here,  viz.  that  in  which  the 
directions  of  the  forces  are  lines  in  one  plane.  Such  forces 
are  called  '4  co-planar."  The  first  step  will  be  to  find,  if 
possible,  a  single  force  which  will  in  every  way  produce 
the  same  effect  as  all  the  forces  acting  together.  Such  a 
force  which  is  equivalent  to  the  combined  action  of  all  the 
individual  forces  is  called  their  "  resultant." 

The  resultant  of  two  forces  must,  then,  produce  the 
same  translation  as  these  two  combined,  and  must  also 
have  the  same  moment  around  any  axis  as  the  combined 
moments  of  the  two  forces.  If  the  resultant  of  two  forces 
can  be  found,  it  can  be  combined  with  a  third  force,  etc. ; 
and  so  the  resultant  of  any  number  of  forces  can  be  found. 

It  will  be  neces- 
sary to  distinguish  ' 
two  cases  :  a,  two 
non-parallel  forces  ; 
b,  two  parallel 
forces. 

53.  Non- parallel 
Forces.  Two  non- 
parallel  forces  lying  R 
in  the  same  plane 
and  represented  by 
the  lines  A  B  and 
CD,  act  on  a  rigid 
body  at  the  points 
B  and  D,  respec- 
tively (e.  g.  two 
strings  are  fastened 
to  nails  at  B  and  D,  and  are  then  pulled  in  the  given 
directions  with  the  specified  forces).  Prolong  the  lines  of 


FIG.  39. 


62 


THEORY   OF  PHYSICS 


CH.  II 


action  until  they  meet  in  a  point,  0.  Let  it  be  in  the  body 
itself.  Since  the  body  is  rigid,  the  force  A  B  will  produce 
the  same  effect  if  applied  at  any  point  in  its  line  of  action, 
e.  g.  at  the  point  0.  Similarly  the  force  CD  may  be  con- 
sidered as  also  applied  at  0.  Since  the  two  forces  may  be 
regarded  as  acting  at  the  point  0,  the  motion  of  transla- 
tion of  0  must  obviously  be  given  by  the  geometrical  sum 
of  the  forces  taken  at  that  point.  That  is,  lay  off  from  0 
two  lines,  0  R  and  0  Q,  equal  in  length  and  direction  to 
A  B  and  CD,  and  add  them.  Their  sum,  OP,  is  then 
equivalent  to  them  so  far  as  translation  is  concerned,  and 
may  be  considered  applied  at  any  point  in  its  line  of  action, 
e.  g.  0  or  0'. 

But  the  force  0  P  will  also  have  the  same  moment 
around  any  axis  as  the  combined  moments  of  its  two  com- 
ponents, A  B  and 
C~D.  This  may  be 
proved  in  the  fol- 
lowing way :  Take 
any  axis  perpendic- 
ular to  the  plane  of 
the  forces,  and  let 
its  intersection  with 
the  plane  be  the 
point  8.  The  mo- 
ment of  0  Q  around 
S  equals  the  pro- 
duct of  0  Q  and  the 
perpendicular  dis- 
tance from  S  to  the  direction  0  Q,  L  e.  S  Si ;  but  this  is 
equal  to  a  number  which  is  twice  the  area  of  the  triangle 
(OSQ).  Similarly,  the  moment  of  0  R  around  S  eanals  a 
number  which  is  twice  the  area  of  the  triangle  (OSJK)', 
but,  in  accordance  with  the  definition  of  Article  .41,  this 
must  be  considered  negative.  Hence  the  combined  mo- 
ment of  0  Q  and  0  R  around  8  is  equal  to  twice  the  dif- 


53]  u        CALIFORNIA  63 

DEPARTMENT  OF  PHYSICS 

ference  in  the  areas  of  the  triangles  (0  S  Q)  and  (0  S  R\ 
But,  by  ordinary  geometry, 


Hence 


-(FOR) 
(GSQ)-(OS£)  = 

=  -(OSP). 


But  —  2  (OSP)  is  the  moment  of  OP  around  S.  Conse- 
quently, the  moment  of  0  P  equals  the  sum  of  the  mo- 
ments of  0  Q  and  0  R,  around  any  axis.  Therefore  the 
force  0~P  is  the  resultant  of  0~Q  and  0jft  ;  that  is,  of  AB 
and  C~D. 

Consequently,  the  resultant  of  two  non-parallel  forces 
which  lie  in  the  same  plane  is  a  force  which  also  lies  in 
this  plane,  whose  numerical  value  and  direction  is  given 
by  the  geometrical  sum  of  the  two  forces,  and  whose  line 
of  action  is  such  as  to  pass  through  the  point  of  intersec- 
tion of  the  two  forces. 

The  same  demonstration  holds  even  when  0,  the  point 
of  intersection  of  the  two  forces,  is  not  a  point  of  the  body. 
For,  as  has  just  been 
proved,  when  the  point  is 
in  the  body,  a  force,  0  P, 
the  geometrical  sum  of  the 
two  forces,  is  the  result- 
ant ;  and  it  may  be  ap- 
plied at  any  point,  0',  in 
the  line  of  action  of  OP. 
But,  now  let  the  portion 
of  the  body  surrounding 
the  point  0  be  removed  ; 
no  change  is  made  in  the 
two  forces,  the  resultant  is 
the  same  ;  and  yet  the  point  of  intersection  is  not  now  a 
point  of  the  body.  As  an  illustration,  let  two  forces,  FI 
and  FZ)  act  on  a  rigid  bar.  Prolong  the  lines  of  action 


64 


THEORY  OF  PHYSICS 


[CH.  II 


until  they  meet  at  0\  lay  off  0  Q  =  ft,  OR  =  F2<  con- 
struct  their  sum,  OP.  Then  the  resultant  of  ft  and  ft 
is  a  force  equal  to  0  Py  applied  at  the  point,  Of,  where  the 
direction  of  the  force  meets  the  bar. 

54.   Parallel  Forces.     Two  parallel  fories,  ft  and  ft,  act 
on  a  rigid  body  at  the  points  A  and  B.  respectively.     To 


ft 


FIG.  42. 


P 
FIG.  43. 


find  their  resultant,  consider  first  the  case  where  the  two 
forces  are  not  quite  parallel :  the  lines  of  action  will  meet 
in  a  point,  0,  at  a  great  distance.  Lay  off  from  this 
point,  0,  two  lines  equal  to  jft  and  ft ;  their  geometrical 
sum  is  the  line  0  P.  But,  in  the  limit,  when  the  lines 
are  parallel,  the  line  OP  equals  the  algebraic  sum  ft 
+  ft,  and  is  parallel  to  the  component  forces.  Hence 
the  resultant  of  two  parallel  forces  lies  in  their  plane, 
is  equal  to  their  algebraic  sum,  and  is  parallel  to  them. 
Further,  it  must  occupy  such  a  position  with  relation 
to  ft  and  ft  that  its  moment  around  any  axis  perpen- 
dicular to  the  plane  is  equal  to  the  sum  of  the  moments  of 


55] 


DYNAMICS 


65 


Fi  and  F2  around  the  same  axis.  Let  Q  be  the  intersection 
of  any  such  axis  with  the  plane  ;  draw  a  perpendicular 
from  Q  to  the  three  parallel  lines*  representing  the  forces, 
intersecting  them  in  the  points  Ri,  R2,  Rz.  Then  the  mo- 
ment of  Fi  around  the  axis  is 
,  (Fl  + 


R*  Q  ;  that  of  Fl  + 
have  such  a  length  that 


X  Ri_Q  ;  that  of  Fz,  F2  X 
F2)  x  £s  Q.    Hence  Rs  Q  must 


Hence        Fl  & 
or 


+ 


.     .     .     (28) 


That  is,  Rz  is  such  a  point  that  the  moment  of  F\  around 
an  axis  through  it  perpendicular  to  the  plane  equals  the 
moment  of  Fz  around  it.  So  the  relative  position  as  well 
as  the  numerical  value  and  direction  of  the  resultant  is 
fixed.  It  may  be  applied  at  any  point  in  its  line  of  action. 
55.  As  a  special  case,  consider  two  particles  whose 
masses  are  mi  and  w2,  and 
which  are  connected  by  a  mass- 
less  beam.  Let  them  be  under 
the  action  of  the  earth ;  the 
mutual  influence  of  the  two 
particles  is  zero  (Art.  32),  but 
there  is  a  force  mig  acting 
on  mi,  and  a  parallel  force 
mzg  acting  on  w2.  The  resul- 
tant, then,  is  (mi  -f  m2)  g  in  a 
parallel  direction,  and  so  placed 
that  FIG.  44. 


mlg 


Its  point  of  application  will  be  where  the  line  intersects 
the  beam.  By  similar  triangles  this  point,  5,  must  be  such 
that 

m1ab  =  ra2  b  c. 
3 


66 


THEORY   OF  PHYSICS 


[CH.  II 


Hence  I  must  be  the  centre  of  inertia  of  the  two  masses ; 
because,  taking  as  the  plane  of  reference  one  perpendicular 
to  the  beam  at  the  pouit  a,  the  position  of  the  centre  of 
inertia,  x,  is  given  by  the  equation, 

(m-L  -f-  ???2)  x  =  mi  •  0  +  w?2  a  c. 
Hence,  calling  a  point  ~b'  the  centre  of  inertia, 

x  =  a  b',   and   (mi  4-  w2)  a  b'  =  m2  a  c. 
Thus  ml  a  b'  =  ra2  (a  c  —  a  b')  =  m2  b'  c. 

Therefore  V  coincides  with  b.  Similarly,  if  there  are  any 
number  of  particles  (or  a  single  large  body)  under  the 
action  of  the  earth,  the  resultant  is  the  entire  mass  multi- 
plied by  g ;  and  its  line  of  action  is  vertically  downward, 
and  passes  through  the  centre  of  inertia. 

56.  The  same  demonstration 
as  given  above  in  Article  54 
holds  good  when  the  forces, 
though  parallel,  are  in  oppo- 
site directions.  The  resultant 
of  the  two  forces,  FI  and  F2, 
is  their  algebraic  sum,  —  that 
is,  the  difference  between  them, 
because  they  are  in  opposite 
directions ;  and  it  must  be  so 
placed  that  its  moment  around 
any  axis  equals  the  sum  of  the 
moments  of  the  two  compo- 
nents around  the  same  axis. 
That  is,  using  F\  and  F3  as 
FIG.  45.  mere  numbers, 


F2 


21 Q 


Hence 


58] 


DYNAMICS 


67 


F2=~FX 


57.  Couples.  Since  the  resultant,  when*  the  two  parallel 
forces  are  in  opposite  directions,  is  FI  —  Fz ;  if  the  forces 
have  equal  numerical  values,  the 
resultant  is  zero.  Such  a  combina- 
tion of  two  equal  parallel  forces 
in  opposite  directions  is  called  a 
"  couple ; "  and  there  is  no  single 
force  which  is  equivalent  to  it. 
The  "  strength  "  of  the  couple  is  a 
name  given  to  the  product  of  the 
numerical  value  of  either  of  the 
forces  and  the  perpendicular  dis- 
tance between  them.  The  moment 
of  the  couple  around  any  axis  per- 
pendicular to  the  plane  of  the  forces 
is 


that  is,  it  equals  the  strength  of  the  couple. 

58.  It  may  be  proved  without  much  difficulty  that  in 
the  most  general  possible  case  of  any  number  of  forces,  in 
one  plane  or  not,  acting  on  a  rigid  body,  the  resulting 
action  is  a  single  force  combined  with  a  single  couple 
in  a  plane  perpendicular  to  the  force. 

As  was  stated  in  Article  50,  when  a  single  force  acts  on 
a  body,  if  its  line  of  action  passes  through  the  centre  of 
inertia,  there  is  only  translation  ;  but  if  its  line  of  action 
does  not  pass  through  the  centre,  there  will  be  translation 
and  also  rotation  around  the  centre  of  inertia  as  if  it  were 
a  fixed  point.  Consequently,  since,  when  any  number  of 
forces  lying  in  one  plane  act  on  a  body,  there  is  a  single 
resultant  force  (except  in  the  case  of  two  equal  and  oppo- 
site parallel  forces),  there  is  only  translation  if  the  result- 
ant passes  through  the  centre  of  inertia ;  otherwise  there 
is  both  translation  and  rotation.  If  there  is  a  couple, 
there  is  no  translation,  only  rotation. 


68 


THEORY   OF  PHYSICS 


[CH.  II 


EQUILIBRIUM 

59.    A  body  or  a  system  is  said  to  be  in  "  equilibrium " 
when  there  is  no  change  in  its  existing  motion ;  that  is, 
-,        when  there  is  no  change  in  its  linear  or 
/        angular  momentum.     The  mathematical 
/         conditions  are,  therefore,  if  there  are  any 
/pa       external  forces,  (1)  the  sum  of  the  com- 
r          ponents  of  the  forces   in  any  direction 
'  must  equal   zero ;    (2)  the  sum   of   the 

/  moments  of  the  forces  around  any  axis 

must  equal  zero. 

Special  Cases.  60.  1.  A  particle  acted 
upon  by  three  forces.  It  was  proved 
(Art.  52)  that,  when  two  forces,  FI  and 
Fz,  acted  upon  a  particle,  their  resultant 
was  their  geometrical  sum.  Therefore, 
if  there  is  a  third  force,  Fs,  acting  on 
the  particle,  which  is  equal  but  opposite 
to  the  resultant,  it  will  exactly  neutral- 
ize the  action  of  F\  and  F2.  Conse- 
quently the  particle  will  be  in  equilibrium. 
The  geometrical  condition,  then,  for  the 
equilibrium  of  a  particle  under  the  action 
of  three  forces,  F^  F2,  F3,  is  that  they 
shall  form  a  closed  triangle  when  added 
geometrically. 

Another  and  better  way  of  considering 
this  case  is  to  form  the  components  of 
the  forces  in  any  direction,  and  to  ex- 
press the  fact  that  the  sum  must  be  zero. 
As  an  illustration  of  the  method,  con- 
sider  the   equilibrium    of    a    system    as 
shown.     B  C  is  a  rigid  beam  which  is 
held   pressed   perpendicularly  against   a 
vertical  wall  under  the  action  of  a  string,  A  B,  fastened  to 


61] 


DYNAMICS 


69 


the  wall  at  A  and  to  the  beam  at  B,  and  a  body  of  mass, 
m,  which  hangs  vertically  below  B.  (The  string  does  not 
slip  over  B,  but  is  fastened  there.)  The  point  B  will  come 


FIG.  48. 

to  rest,  and  is  therefore  in  equilibrium ;  and  three  forces 
are  acting  there  :  (1)  the  force  m  g,  due  to  the  earth,  verti- 
cally down  ;  (2)  the  tension  T  of  the  string  in  the  direc- 
tion B  to  A ;  (3)  the  reaction  R  of  the  wall  horizontally 
outward.  (The  beam  is  pressed  against  the  wall,  therefore 
there  is  an  equal  reaction  of  the  wall  against  the  beam.) 
Call  the  angle  A  B  C,  0.  Then  the  sum  of  the  components, 
vertical  and  horizontal,  are 

T  sin  6  -  mg     and     T  cos  0  -  R, 
and  these  must  separately  equal  0  ;  i.  e. 

TsinO  —  mg  =  0, 
-  R  =  0. 


These  equations  may  be  verified  by  direct  experiment,  as 
all  the  quantities  can  be  measured  separately. 

61.  2.  A  large  rigid  body  acted  upon  by  three  forces. 
It  was  proved  in  Articles  53  and  54,  that  when  two  forces 
acted  on  a  body  they  had  a  resultant  if  they  were  in  the 


70 


THEORY  OF  PHYSICS 


[CH.  II 


same  plane  and  if  they  did  not  form  a  couple.  Conse- 
quently, if  the  third  force  is  equal  and  o'pposite  to  the 
resultant  of  the  other  two,  and  if  it  is  applied  in  the  same 
line  as  their  resultant,  there  will  be  equilibrium.  The 
two  cases  of  non-parallel  and  parallel  forces  must  be 
distinguished. 

62.  a.  Three  Non-parallel  Forces.  The  resultant  of  two 
forces,  FI  and  F2,  lying  in  one  plane  is  found,  as  explained 

in  Article  53,  by  prolong- 
ing their  lines  of  action 
until  they  meet,  and  con- 
structing at,  that  point 
their  geometrical  sum. 
A  force,  F3,  then,  equal 
and  opposite  to  the  re- 
sultant, must  lie  in  the 
same  plane  as  FI  and  Fz ; 
its  line  of  action  must 
pass  through  the  point  of 
intersection  of  the  lines 
of  action  of  F\  and  F2 ; 
and  its  numerical  value 
must  be  the  same  as  that 
and  F2>  and  it  must  be  in  the  oppo- 
site direction.  Hence,  if  a  body  is  in  equilibrium  under 
the  action  of  three  non-parallel  forces,  they  must  all  lie 
in  the  same  plane,  their  lines  of  action  must  meet  in  the 
same  point,  the  sum  of  their  components  in  any  direction 
must  equal  zero. 

As  an  illustration,  consider  the  case  of  a  beam,  A  B,  at 
rest  lying  on  two  smooth  inclined  planes.  There  are  three 
forces  acting  on  the  beam  :  (1)  the  force  m  g  vertically  down 
through  the  centre  of  inertia :  (2)  the  reaction  El  of  the 
inclined  plane  at  A,  which  is  perpendicular  to  the  plane 
since  it  is  smooth  ;  (3)  the  reaction  R%  of  the  second  inclined 
plane  at  B.  These  forces  must  all  lie  in  a  plane,  meet  in 


FIG.  49. 


of  the  resultant  of 


63] 


DYNAMICS 


71 


FIG.  50. 

a  point  0,  and  neutralize  one  another,  since  the  beam  is  at 

rest. 

63.   b.  Three  Parallel  Forces.     As  explained  in  Articles 

54  and  55,  if  there  is  a  resultant  of  two  parellel  forces,  it  is 

a  third  force  which  lies  in  their  plane, 

which  is  parallel  to  them,  and  whose  nu- 
merical value  equals  their  algebraic  sum  ; 

and  it  is  so  placed  relatively  to  them  that 

its  moment  around  any  axis  perpendicular 

to  their  plane  equals  the  sum  of  their  mo- 
ments around  the  same  axis.     Therefore, 

if  three  parallel  forces,  Fly  F2)  Fz,  are  in 

equilibrium,  the  algebraic  sum  F\  +  F2  + 

Fs  must  equal  zero,  and  the  sum  of  the 

moments  around  any  axis  must  equal  zero. 

As  an  illus- 
tration, con- 
sider a  rigid 
beam  which 

carries  two  bodies  whose 
masses  are  m\  and  mz,  and 
which  is  itself  suspended  by 
a  string  so  placed  that  the 

mi  7/</2     system  is  at  rest.    (The  beam 

FIG.  52.  J 

is  not  necessarily  horizontal.) 

Let  the  mass  of  the  beam  be  so  small  that  it  may  be 


i 

k  Xj—  ><       Xj-^ 


72 


THEORY  OF  PHYSICS 


[CH.  II 


neglected.  There  are,  then,  three  parallel  forces  acting  on 
the  beam :  T,  the  tension  of  the  string  up ;  and  MI  g  and 
mz  g  down.  They  are  all  in  the  same  plane ;  and  since  the 
system  is  at  rest, 

T  —  mi  9  —  mz  9  =  0. 

Further,  the  points  of  application  must, be  such  that,  tak- 
ing moments  around  any  axis  perpendicular  to  the  plane, 
the  sum  will  be  zero.  Take  them  around  an  axis  perpen- 
dicular to  the  plane  of  the  forces  at  the  point  where  the 
tension  T  is  applied.  The  moment  of  T  is  zero ;  hence 

=  0. 


These  two  conditions  may  be  verified  by  experiment. 

64.    3.  A  body  acted  upon  by  two  couples.    It  was  proved, 
in  Article  57,  that  a   couple   composed   of  two  parallel 

forces,  each  equal  to 
F,  whose  distance  apart 
was  I,  i.  e.  of  strength 
Fl,  produced  a  mo- 
ment Fl  around  any 
axis  perpendicular  to 
•po  their  plane.  A  couple 
has  no  resultant ;  and 
consequently  its  ac- 
tion can  only  be  neu- 
tralized by  another  cou- 
ple of  equal  strength 
but  opposite  to  it. 
Thus,  if  two  couples 
whose  strengths  are 
FI  h  and  F2 12  act  on  a 
body,  and  if  it  is  in  equilibrium,  they  must  lie  in  the  same 
plane,  and  their  strengths  must  have  equal  numerical  val- 
ues ;  but  the  moments  must  be  in  opposite  directions.  In 
the  diagram  the  couple  FI  li  tends  to  produce  rotation  in 


65]  DYNAMICS  73 

the  direction  opposite  to  the  hands  of  a  watch,  while  F*  12 
tends  to  produce  rotation  in  the  same  direction  as  that  of 
the  hands.  Consequently,  if  the  body  is  in  equilibrium, 
the  numerical  values  are  equal,  i.  e. 


(29) 


65.  Stability  of  Equilibrium.  Even  though  two  bodies 
or  systems  are  in  equilibrium,  there  may  be  differences  in 
their  conditions.  Thus  a  cone  balanced  on  its  point  is  in 
equilibrium,  as  it  is  also  when  it  is  lying  on  its  side  or 
resting  on  its  base  ;  but  there  are  obvious  differences  be- 
tween the  three  states.  Similarly  an  ellipsoid  set  in  rota- 
tion around  its  longest  axis  may  be  in  equilibrium,  as  it 
may  be  also  when  in  rotation  around  its  axis  of  intermedi- 
ate length  ;  but  its  properties  are  quite  different  in  the  two 
cases.  The  simplest  way  of  considering  these  points  of 
difference  is  to  note  what  happens  when  the  body  or 
system  is  given  a  slight  displacement  from  its  condition 
of  equilibrium. 

Equilibrium  is  said  to  be  "  stable,"  if  when  the  displace- 
ment occurs,  the  change  produced  becomes  immediately 
less  and  less,  and  is  finally  reversed  ;  so  that  the  body  or 
system  returns  towards  its  previous  condition.  In  general, 
the  system  will  then  oscillate  through  the  position  of  equi- 
librium. Thus,  an  ordinary  pendulum  when  hanging  at 
rest  is  in  stable  equilibrium  ;  because,  when  it  is  struck  a 
slight  blow,  the  resulting  velocity  becomes  less,  and  is 
finally  reversed  ;  and  then  the  pendulum  makes  oscilla- 
tions. Similarly,  an  ellipsoid  spinning  about  its  greatest 
or  least  axis  is  in  stable  equilibrium.  These  are  illus- 
trations of  what  are  respectively  called  "  statical  "  and 
to  kinetic  "  stability. 

Equilibrium  is  said  to  be  "  unstable,"  if,  when  the  system 
is  displaced,  the  initial  change  goes  on  increasing.  Thus, 
a  cone  balanced  on  its  apex  is  unstable,  because,  if  it  is 
given  a  blow,  the  initial  velocity  will  increase.  Similarly, 


74  THEORY  OF  PHYSICS  [CH.  II 

an  ellipsoid  spinning  around  its  intermediate  axis  is  un- 
stable, because,  if  displaced,  it  will  tend  to  spin  around  its 
greatest  or  its  least  axis. 

Equilibrium  is  said  to  be  "  neutral,"  if,  when  the  system 
is  displaced,  the  initial  change  becomes  permanent,  and 
neither  increases  nor  decreases.  A  sphere  resting  on  a 
smooth  table  is  in  neutral  equilibrium ;  because,  if  it  is 
given  a  blow,  the  velocity  thus  produced  will  not  be 
changed ;  and  the  sphere  will  move  in  a  straight  line  with 
a  constant  speed. 

66.  Principle  of  Stable  Equilibrium.  The  most  common 
condition  of  equilibrium  in  nature  is  stability ;  because,  of 
course,  slight  changes  are  always  occurring;  and  so  all 
bodies  tend  to  pass  into  stable  conditions.  Any  disturb- 
ance of  stability  must  produce  a  reaction  which  tends  to 
restore  the  body  or  system  to  its  previous  condition ;  and 
this  principle  can  be  applied  to  any  stable  condition, 
whether  it  is  a  purely  mechanical  one  or  not.  Consider 
some  illustrations  of  stability.  1.  A  body  hanging  sus- 
pended by  a  spiral  spring  is  in  stable  equilibrium.  If  a 
blow  downward  is  given  it,  the  initial  velocity  will  be 
decreased,  owing  to  the  increased  tension  of  the  spring. 
Hence,  if  the  tension  of  a  stretched  spiral  spring  is  in- 
creased by  any  means,  it  will  raise  the  suspended  body. 
2.  An  iron  bar  surrounded  by  some  medium,  e.  g.  water,  at 
a  constant  temperature  is  in  stable  equilibrium ;  for  if  its 
temperature  is  suddenly  increased  in  any  way,  the  tendency 
will  be  for  it  to  return  to  the  temperature  of  the  surrounding 
medium.  Now,  when  the  temperature  of  an  iron  bar  is 
increased,  its  length  is  increased ;  but  this  act  of  increas- 
ing in  length  produces  a  tendency  for  the  bar  to  return  to 
its  former  temperature.  That  is,  if  an  iron  bar  is  stretched 
by  mechanical  means,  its  temperature  will  fall.  3.  Just 
the  opposite  effect  happens  with  a  piece  of  rubber  cord. 
When  its  temperature  is  raised,  it  shrinks ;  consequently 
compressing  the  rubber  diminishes  its  temperature,  and 
stretching  it  raises  its  temperature. 


67]  DYNAMICS  75 

ENERGY 

67.  Work  and  Energy.  The  words  "  to  do  work  "  convey 
a  more  or  less  definite  idea  to  every  one,  and  any  body 
that  has  the  power  to  do  work  is  said  to  have  "  energy." 
Energy  may  be  defined,  then,  as  a  measure  of  the  possi-i 
bility  of  doing  work.  Consider  a  few  familiar  illustrations. 
If  a  body  is  raised  vertically  from  the  earth,  work  is  done 
by  whatever  raises  it ;  if,  for  instance,  a  compressed  spiral 
spring  is  allowed  to  expand  and  thus  raise  the  body,  the 
spring  had  energy  when  it  was  compressed  and  loses  it 
when  it  is  expanded  Work  is  done  whenever  the  speed 
of  a  body  is  increased ;  thus,  if  a  bullet  is  fired  from  a  rifle, 
the  powder  has  energy  before  it  is  exploded,  and  then  loses 
it  in  the  act  of  giving  the  bullet  its  motion.  Work  is 
required  to  make  the  fly-wheel  of  a  steam-engine  revolve 
faster ;  the  steam  in  the  cylinder  has  the  energy,  and  then 
loses  it.  Work  is  done  when  a  clock-spring  is  wound  up, 
when  a  spiral  spring  is  compressed,  or  when  a  gas  is  con- 
densed. It  is  to  be  particularly  noted  that  whenever  work 
has  been  done  upon  any  body,  it  itself  is  given  the  power 
of  doing  work ;  that  is,  it  itself  has  energy  given  it  as  the 
result  of  another  body  losing  energy.  Thus,  the  body 
raised  vertically  above  the  earth  can  do  work  when  it  falls  ; 
for  instance,  it  may  compress  the  spiral  spring  again  ;  the 
bullet  in  motion  can  also  compress  a  spring,  thus  doing 
work  and  losing  energy;  the  fly-wheel  in  rotation  has 
energy,  which  can  be  used  in  winding  up  a  spring  or  in 
working  a  pump ;  a  compressed  gas  has  energy  also,  be- 
cause it  can  do  work  by  expanding.  Consequently  it  may 
be  said,  in  general,  that  work  is  done  as  a  result  of  one 
body  losing  energy  and  another  gaining  it;  and  experi- 
ments most  carefully  performed  seem  to  prove  definitely 
that  the  amount  of  energy  lost  by  one  body  exactly  equals 
that  gained  by  the  second.  This  means  that  starting  with 
a  certain  amount  of  energy  it  is  impossible  by  any 


76  THEORY  OF  PHYSICS  [CH.  II 

machinery  or  process  to  get  more  than  a  definite  amount 
of  work  done  by  any  one  transformation. 

Work  is  done,  of  course,  in  many  other  ways  than  those 
mentioned  in  the  above  paragraph.  Thus,  work  is  done 
when  a  piece  of  iron  is  pulled  away  from  a  fixed  magnet ; 
and  if  the  iron  is  allowed  to  move  back  to  the  magnet, 
work  may  be  done  by  it.  Again,  work  is  done  whenever 
one  portion  of  matter  is  made  to  move  over  another,  as 
two  boards  rubbed  together,  a  jet  of  water  discharged  into 
water  or  air,  etc.  And,  as  a  result  of  this  so-called  "  fric- 
tional "  work  being  done,  it  is  noticed  that  the  temperature 
of  the  bodies  rises,  or  some  other  so-called  "  effect  of  heat " 
is  produced.  It  is  possible,  though,  by  making  use  of  dif- 
ferences of  temperature,  to  do  work,  as  is  shown  by  a 
steam-engine  ;  and  consequently,  in  this  case  too,  energy 
is  given  the  bodies  as  a  result  of  the  work  done.  It  will 
be  shown  later  on  that  all  the  "  effects  of  heat "  are  in 
reality  due  to  changes  in  the  energy  of  the  smallest 
portions  of  the  matter  in  a  body,  not  the  energy  of  the 
•body  as  a  whole.  A  burning  gas  has  energy,  because  it 
can  do  work ;  so  has  a  Leyden  jar  "  charged  "  with  elec- 
tricity ;  so  has  a  Voltaic  cell  which  produces  an  electric 
current ;  and  so  on  indefinitely. 

But  to  all  these  manifestations  of  energy  the  same  law, 
as  above  stated,  applies :  the  energy  lost  by  one  body  or 
system  is  exactly  equal  to  that  gained  by  another.  This 
principle  has  been  confirmed  by  experiment  in  numerous 
ways ;  every  observed  phenomenon  in  nature  is  in  perfect 
accord  with  it ;  and  so  it  is  regarded  as  one  of  nature's 
great  laws.  It  is  sometimes  called  the  "  Principle  of  the 
Conservation  of  Energy." 

68.  Measure  of  Work  and  Energy.  It  should  be  noticed 
that,  in  every  case  of  work  being  done,  only  two  ideas  are 
involved :  motion  of  a  certain  amount,  and  what  has 
been  called  a  force.  When  a  body  is  raised  vertically 
from  the  earth,  there  is  a  force  downward  "  acting  on  "  it 


68]  DYNAMICS  77 


to  mg  where  m  is  its  mass  and  g  is  a  certain  con- 
stant, as  is  proved  by  experiment  (Art.  33)  ;  and  the 
amount  of  work  done  depends  upon  the  numerical  value  of 
this  force  and  upon  the  height  though  which  the  body  is 
raised.  If  a  spiral  spring  is  compressed,  there  is  always  a 
force  acting  in  the  opposite  direction  ;  and  the  amount  of 
work  done  depends  upon  this  force  and  the  distance 
through  which  the  spring  is  compressed.  It  is  a  question 
of  distance,  not  time  ;  because,  for  instance,  when  a  spiral 
spring  is  compressed,  the  amount  of  energy  it  has  cannot 
depend  upon  the  time  taken  to  compress  it.  And  if  there 
is  no  motion,  no  work  is  done.  A  spring  kept  compressed 
does  no  work  ;  nor  does  a  pillar  supporting  a  building. 
When  a  body  has  its  speed  changed,  there  is  by  definition 
a  force;  and  the  longer  the  distance  through  which  the 
force  acts,  the  greater  is  the  change. 

The  motion  upon  which  depends  the  amount  of  work 
done  is  the  distance  traversed  in  the  direction  of  the  force  , 
because,  if  a  body  moves  at  right 
angles  to  a  force,  no  work  is  done. 
Thus,  if  a  ball  rolls  along  a  hori- 
zontal table,  no  work  is  done  under 
the  action  of  the  earth  or  against 
it.  And  if  a  body  is  moved  in  a 
direction,  A  B,  which  is  inclined 
to  that  of  the  force  F,  the  work 
done  depends  on  the  distance  A  C,  FIG.  54. 

the  component  of  A  B  in  the  line 

of  action  of  the  force;  because  the  motion  from  A  to  £ 
can  be  accomplished  by  going  from  A  to  C  and  from  C 
to  B\  and  in  this  last  portion  no  work  is  done.  Since, 
then,  when  work  is  done,  the  motion  must  be  in  the 
direction  of  the  force,  if  the  work  is  done  in  producing 
change  of  motion  of  a  body,  the  change  produced  must 
be  in  the  speed,  not  in  the  direction  of  motion.  Thus, 
when  a  body  is  moving  in  a  horizontal  circle  with  a 


78  THEORY   OF   PHYSICS  [CH.  II 

constant  speed,  no  work  is  being  done,  and  the  energy 
remains  constant. 

The  amount  of  work  done  depends  upon  force  and  dis- 
tance in  the  direction  of  the  force ;  consequently  the 
numerical  value  of  the  work  done  may  be  defined  as 
the  product  of  the  numerical  values  of  the  force  and  the 
distance  in  the  direction  of  the  force  through  which  the 
motion  takes  place  while  the  force  is  acting.  The  unit  of 
work  is,  then,  the  product  of  one  dyne  and  one  centimetre, 
and  is  called  the  "  erg."  Since  amount  of  energy  is  meas- 
ured by  the  amount  of  work  done  or  which  can  be  done, 
the  erg  is  also  the  unit  of  energy.  Thus,  if  a  body  whose 

mass    is    m    grams 

B  ' is   raised  vertically 

through  a  height 
h  centimetres,  the 
force  overcome  has 
been  m  g  dynes  ; 
and  the  work  done 


A      FX'O.  55.  is  mg  h  ergs.     This 

work  depends  sim- 
ply on  the  vertical  distance  h;  being  entirely  independent 
of  the  path  itself.  For  if  there  are  two  horizontal  planes  a 
distance  h  apart,  the  same  amount  of  work  is  done  in  rais- 
ing a  given  body  from  any  point  of  the  lower  plane  to  any 
point  of  the  upper  along  any  path.  Let  A  B  be  a  vertical 
line,  A'  B'  an  inclined  one,  and  A"  B"  a  curved  one.  The 
work  done  along  A'  B1  has  just  been  proved  to  be  the  same 
as  that  done  along  A  B ;  and  since  any  curved  line  may 
oe  considered  as  the  limiting  form  of  a  broken  line  con- 
sisting of  a  great  number  of  extremely  short  straight, 
lines,  the  work  done  along  it  is  also  the  same.  Since 
this  work,  m  g  h  ergs,  has  been  done  on  the  body,  its 
energy  has  been  increased  by  this  much,  owing  to  its 
change  of  position  with  reference  to  the  earth.  (If  its 
final  speed  is  different  from  its  initial,  there  will  be  an 
additional  change  in  its  energy.) 


68] 


DYNAMICS 


79 


Again,  consider  a  vertical  spiral  spring  resting  on  a 
horizontal  platform,  and  let  it  be  compressed  in  some  way, 
e.  g.  by  means  of  one's  hand.  Work  is  done, 
because  a  force  is  overcome  ;  and  the  spring 
gains  as  much  energy  as  is  lost  by  the  body 
doing  the  work. 

No  idea  can  be  formed  at  present  of  what 
the  nature  of  the  energy  in  these  last  two 
illustrations  is  ;  because  it  is  not  expressed 
in  terms  of  matter  and  motion.     (See  Art. 
7.)     And  such  forms  of  energy  as  depend 
upon  the  positions  of  the  bodies  or  upon 
the  configuration  of  a  body  or  system  are     E 
called  "  potential  energy."     Thus  any  body          FlG  56 
has  potential  energy  with  reference  to  the 
earth  ;  a  compressed  spring  has  potential  energy  ;  so  has 
a  twisted  wire,  a  piece  of  iron  near  a  magnet,  a  charged 
Leyden  jar,  etc.     Ultimately,  of  course,  it  may  be  possible 
to  describe  any  form  of  potential  energy  in  terms  of  mo- 
tion of  matter,  when  the  phenomenon  comes  to  be  better 
understood.     So  "  potential  energy  "  must  be  understood  to 
be  a  term  which  is  used  to  cover  our  present  ignorance. 

When  work  is  done  in  changing  the  speed  of  a  body, 
then,  not  alone  is  the  amount  of  work  known,  but  also  the 
nature  of  the  energy.  Thus,  let  a  body  whose  mass  is  m 
have  its  linear  speed  changed  from  s0  to  s,  the  direction  re- 
maining constant.  By  Formula  (3),  Article  18, 

2  a  x  =  s2  —  s02, 

where  a  is  the  acceleration  and  x  the  distance  required  for 
the  change.  By  definition,  the  force  is  ma\  hence  the 
work  is 

02     .     .     .     .     (30) 


m  a  x  =     m  s   — 


m 


Consequently  the  energy  has  been  increased  by  this  amount. 
If  the  body  had  been  at  rest  initially,  i.  e.  if  SQ  =  o,  it 


80  THEORY  OF  PHYSICS  [CH.  II 

would  have  had  no  energy  of  motion  then  ;  but,  after  the 
work  |  m  s2  was  done,  it  would  have  had  this  amount  of 
energy  due  to  its  linear  speed,  s.  Therefore,  when  a  body 
whose  mass  is  m  has  a  linear  speed  s,  it  has  an  amount  of 
energy,  |  ms2,  owing  to  its  motion.  This  energy  which  a 
body  has  in  virtue  of  its  being  in  motion  is  called  "ki* 
netic  energy."  A  special  illustration  of  kinetic  energy  is  a 
body  in  rotation  around  a  fixed  axis.  Consider  the  sim- 
plest case,  that  of  a  particle  whose 
mass  is  m  revolving  around  a  point 
0  in  a  circle  of  radius  r  with  a 
constant  angular  speed  «.  Its 
linear  speed,  s,  —  r  CD.  Hence  its 
kinetic  energy  is  |-  mr2o>2.  But 
m  r2  =  /,  the  moment  of  inertia  of 
m  around  0  (Art.  43).  Conse- 
quently, the  kinetic  energy  is  -|-  7&>2. 
FJG  57  (This  is  perfectly  analogous  to  the 

expression   |-  m  s2  for  translation.) 

Let  a  force,  F,  act  on  m,  perpendicularly  to  the  radius  r. 
The  work  done  is  F '  x.  Let  x  be  a  very  small  distance  in 
the  direction  of  the  motion  of  the  particle;  and  let  the 
corresponding  angle  be  0.  Then  x  =  r  0 ;  and  the  work 
Fx=  Fr6  =  L0,  where  L  is  the  moment  around  0.  This 
again  is  perfectly  analogous  to  Fx  for  translation. 

69.  Transfer  of  Energy.  Work  may  be  done  by  the  pas- 
sage of  potential  energy  into  potential,  as  when  a  com- 
pressed spring  is  made  to  raise  a  body  from  the  earth  ;  by 
kinetic  energy  passing  into  kinetic,  as  when  a  ball  in 
motion  strikes  another  ball  and  sets  it  in  motion  ;  by 
kinetic  energy  passing  into  potential,  or  vice  versa.  This 
last  form  of  work  is  by  far  the  most  common.  A  body  at 
a  distance  above  the  earth  has  potential  energy ;  but  if 
allowed  to  fall  freely,  it  loses  potential  energy  and  gains 
kinetic ;  and  of  course  the  loss  of  one  equals  the  gain  of 
the  other.  Thus,  if  a  body  whose  mass  is  m  falls  from  any 


69] 


DYNAMICS 


81 


point  in  one  horizontal  plane  to  any  point  in  another  hori- 
zontal plane  at  a  vertical  depth  h  below  it,  e.  g.  from  B  to 
A  or  Br  to  A'  or  B" 
to  A",  it  will  lose    _  B 

an    amount   of    po- 
tential energy,  mg  h. 
Consequently,  if  the 
motion   is   along   a 
smooth  path,  it  will      — 
gain       an       equal 
amount    of    kinetic 
energy.     That  is,  if  s0  is  its  speed  at  the  upper  plane,  and 
s  that  at  the  lower, 


A' 
FIG.  58. 


A" 


m  s' 


m  s02  =  m  g  h, 


or 


which   is   the  formula   previously  proved  in   Article   18. 

This  proves,  then,  that  no  matter  how  the  body  falls 

along  a  smooth  path  through 
a  given  vertical  height,  the 
change  in  the  square  of  the 
speed  is  the  same.  A  simple 
application  of  this  principle 
is  afforded  when  a  particle 
falls  along  the  inside  of  a 
vertical  circle,  or,  what  is  the 
same  thing,  when  a  particle 
suspended  by  a  cord  from  a 
fixed  point  is  allowed  to 

swing  in  a  vertical  plane.     If  the  particle  moves  freely 

from  Pi  downward  along  the  circle,  starting    from   rest, 

its  speed  at  the  bottom  is  such  that 


But  by  geometry  it  may  be  proved  that 


82  THEORY   OF  PHYSICS  [CH.  II 


Q1A=P1A  /2r, 
where  Pl  A  is  the  chord,  and  r  is  the  radius. 


Hence  Sl=PA\-  ......     (31) 

If  the  particle  had  moved  from  P2  down  the  circle,  its 
speed  at  the  bottom  would  have  been 


This  gives  at  once  a  method  by  which  the  speed 
may  be  measured  in  the  experiment  described  in  Arti- 
cle 29. 

It  is  to  be  noted  that  a  body  left  free  to  fall  will  always 
lose  potential  energy.  Similarly  a  compressed  spring  free 
to  move  loses  potential  energy.  A  piece  of  iron  separated 
from  a  fixed  magnet  has  potential  energy,  but,  if  it  is  free 
to  move,  it  loses  it  and  gains  kinetic  energy.  So  in  all 
cases.  And  it  may  be  regarded  as  a  general  law  that 
there  is  a  constant  tendency  in  any  system  for  its  poten- 
tial energy  to  decrease.  In  fact,  when  in  ordinary  lan- 
guage it  is  said  that  there  is  a  "  force  acting "  between 
bodies,  either  an  "  attraction  "  or  a  "  repulsion,"  all  that 
is  meant  is  that,  if  the  bodies  are  left  free  to  move, 
the  motion  will  be  such  that  the  potential  energy  will 
decrease. 

Work  may  be  done  at  different  rates,  some  bodies  work- 
ing rapidly,  others  slowly ;  and  the  rate  of  doing  work  — 
that  is,  the  amount  of  work  done  in  one  second,  if  the 
rate  is  uniform  —  is  called  the  "  activity  "  or  "  power."  The 
unit  of  activity  is  one  erg  per  second,  and  107  ergs  per  sec- 
ond is  called  a  "  watt."  The  standard  sometimes  used 
commercially  is  the  "horse-power,"  which  is  the  power 
required  to  raise  550  pounds  through  a  vertical  height 


71]  DYNAMICS  83 

of  one  foot  in  one  second.  One  pound  is  453.6  grams, 
and  one  foot  is  30.48  centimetres.  Thus  the  force  is 
550  X  453.6  X  g  dynes;  and  the  work  done  is  550  x 
453.6  X  g  x  30.48  ergs.  But  this  work  is  done  in  one 
second  ;  hence 

550  x  453.6  x  g  x  30.48 
1  horse-power  =  -  —  in7  "  wa^s 

=  746  watts. 

70.  Machines.  Machines  are  mechanisms  by  means  of 
which  work  is  done,  that  is,  energy  transferred  from  one 
body  to  another.  They  themselves  do  no  work  ;  they 
neither  gain  nor  lose  energy,  but  simply  transfer  it.  There 
are  in  most  machines,  however,  parts  which  move  over 
each  other  ;  and  so  some  energy  will  always  pass  into  these 
parts,  producing  "heat-effects"  (see  Art.  67).  A  perfect 
machine  would  be  one  which  gave  out  again  all  the  energy 
given  it,  without  any  "  heat-effect."  Of  course  the  machine 
cannot  deliver  more  energy  than  is  furnished  it  ;  for  this 
would  be  a  violation  of  the  principle  of  the  conservation 
of  energy.  But  the  force  which  is  doing  the  work  need 
not  equal  the  force  produced  ;  for  let  the  work  done  on 
the  machine  be  F\  a?i,  and  that  done  by  the  machine  be 
F2  x*.  Then 


i.  e.  Fl/F2  =  aja/ai!  ;  and  so  they  may  be  different.  This 
ratio  F2/Fi  of  the  force  produced  to  the  force  working  is 
called  the  "  mechanical  advantage  "  of  the  machine.  The 
most  common  forms  of  machines  are  the  lever,  the  pulley. 
and  the  screw. 

71.  The  lever  is  simply  a  rigid  beam  pivoted  at  one 
point.  Thus  let  a  horizontal  beam  rest  on  an  edge  at  0, 
and  let  a  force,  F2,  be  applied  at  a  point,  A*  If  another 
force,  Fit  is  acting  at  another  point,  A\,  the  problem  is  to 


84  THEORY  OF  PHYSICS  [CH.  II 

find  what  the  numerical  value  of  FI  is  when  it  can  just 
overcome  F2  acting  at  A2  (e.  g.  a  man  pulling  down  at  AI 
wishes  to  raise  a  body  hanging  from  A2).  Let  the  forces 
be  parallel,  and  call  the  distances  0  AI  =  ll}  0  A2  =  12. 
Then,  if  the  beam  is  turned  slightly  around  its  pivot 

through  an  angle,  9,  the 
point  AI  moves  through  a 
distance  Xi  =  ^  6  in  the 
direction  of  FI  ;  and  Az 
moves  a  distance  x2  =  12  9 
FIG.  60.  in  the  direction  opposite 

to  F». 


A! 

1 


Hence,  since  Fl  x\  =  F2  x2, 

Fll1  =  FJ,  .     .    .     .     .     .     (32) 

That  is,  if  Fl  and  F2  are  in  this  ratio,  they  will  just  balance 
each  other.  If  their  ratio  is  different  from  this  value,  there 
will  be  acceleration. 

Another  way  of  solving  the  same  problem  is  to  consider 
it  one  of  equilibrium,  and  deduce  the  condition  that  there 
shall  be  no  moment.  There  are  three  forces  acting :  FI, 
Fz,  and  the  resistance  of  the  pivot.  In  order  to  leave  out 
this  last,  it  is  only  necessary  to  take  moments  around  the 
pivot ;  for  then  there  are  only  two  moments,  F1  li  and 
—  F2  /2.  Hence,  rf  there  is  equilibrium,  F\  l\  —  F212  —  0.  or 

JYfceJftJ* 

If  the  forces  are  not  perpendicular  to  the  lever,  or  if 
they  are  not  parallel,  the  same  demonstration  holds  if  I  is 
the  perpendicular  distance  from  the  pivot  to  the  line  of 
action  of  the  force. 

Illustrations  of  levers  are  extremely  common  in  ma- 
chinery and  in  apparatus.  One  of  its  most  important  uses 
is  in  a  so-called  "  chemical  balance."  In  this  apparatus  a 
uniform  horizontal  beam  is  pivoted  at  its  middle  point, 
and  carries  suspended  at  its  ends  from  two  other  pivots  or 


72] 


DYNAMICS 


85 


knife-edges  two  pans  of  equal  masses.  It  is  so  arranged 
that  its  position  of  equilibrium  is  stable.  If  a  body  is 
now  placed  in  each  pan,  there  will  be  in  general  a  move- 


FIG.  61. 


ment  of  the  balance ;  but  if  the  two  forces  produced  by 
the  earth  on  the  two  bodies  are  the  same,  the  balance  will 
not  move  ;  or,  if  displaced,  it  will  make  oscillations  about 
its  position  of  equilibrium.  A  balance  can  be  used,  then, 
to  compare  forces,  and  in  particular  to  compare  weights  of 
bodies  with  reference  to  the  earth. 

72.  A  pulley  is  a  grooved  circular  wheel  free  to  turn 
around  an  axle  passing  through  its  centre.  A  cord  is 
passed  along  the  groove,  so  that  a 
force  applied  to  one  end  of  the 
cord  has  the  direction  of  its  line 
of  action  changed.  The  amount 
of  the  force  is  not  altered,  if  there 
is  no  friction ;  because,  if  F\  and 
F2  balance  each  other,  their  mo- 
ments around  the  axis  are  F\  r 
and  —  F2  r.  Consequently,  if  there 
is  equilibrium, 


,  =  Fz. 


86 


THEORY  OF  PHYSICS 


[CH,  II 


Let  the  cord  pass  over  a  fixed  pulley,  then  over  a  mov- 
able one,  and  back  to  a  fixed  support.  Consider  what 
force  Fi  applied  to  the  free  end  of  the  cord  can  balance 
the  force  Fz,  acting  downward  on  the  movable  pulley, 
when  the  two  portions  of  the  cord  are  parallel.  Since  a 

pulley  simply  changes  the  di- 
rection of  a  force,  the  force 
acting  in  each  portion  of  the 
cord  on  the  two  sides  of  the 
movable  pulley  is  Fl  acting 
up.  Hence,  if  the  movable 
pulley  is  in  equilibrium, 

2  Fl  -  F2  =  0,   or   2  Fl  =  F2. 


In  actual  practice,  F2  must  be 
made  to  include  the  action  of 
the  earth  on  the  pulley ;  and 
a  correction  must  be  made  for 
all  friction. 

Another  way  of  solving  the 
question  is  to  consider  a  small 
displacement  such  that  the 
pulley  is  raised  a  distance  x\ 
that  is,  the  string  is  shortened 

by  an  amount  2  x.     Hence  the  work  done  by  FI  is  Fi  2  x, 

and  that  done  on  F2  is  F2  x,  and  so 


63. 


or 


2  Fl  =  F2. 


If  the  portions  of  the  cord  are  not  parallel,  the  compo- 
nents of  the  forces  in  a  vertical  direction  balance  the  force 
down.  If  acceleration  is  produced,  then,  of  course,  Fl  must 
be  accordingly  greater  than  F2  /  2. 

Pulleys  are  often  made  with  more  than  one  grooved 
wheel  on  the  same  axle.  If  the  wheels  can  rotate  inde- 
pendently of  each  other,  they  are  called  double,  triple,  etc. 
pulleys. 


72] 


DYNAMICS 


87 


Consider  an  arrangement,  as  shown,  where  a  cord  passes 
over  one  wheel  of  a  fixed  double  pulley,  then  over  a  mov- 
able single  pulley,  over  the  second  wheel  of  the  double 
pulley,  and  back  to  the  axle  of  the  movable  pulley. 
There  are  thus  three  portions  of  the  cord  acting  on  the 
movable  pulley,  and  they 
can  be  considered  parallel. 
Consequently,  as  above,  it  is 
evident  that 

3  F1  =  F2. 

In  general,  then,  if  there  are 
n  portions  of  the  same  cord 
pulling  up  the  lower  pulley, 

nF,  =  Ff   .     (33) 

Other  combinations  of  pul- 
leys than  these  with  only 
one  cord  are  possible  ;  but, 
however  many  pulleys  or 
separate  cords  are  used,  the 
mechanical  advantage  may 
always  be  calculated  in  a 
manner  analogous  to  the  one 
just  given. 

Some  pulleys  are  made  with  two  wheels  on  the  same 
axle,  but  clamped  rigidly  together ;  the  grooves  are  not 
smooth,  but  are  cut  into  openings  so  as  to  receive  the  links 
of  a  chain  which  passes  over  them ;  and  one  wheel  has,  as 
a  rule,  one  less  opening  than  the  other,  and  so  has  a 
smaller  radius.  Such  a  pulley  is  called  a  "differential" 
one.  The  chain  is  arranged  as  shown.  It  is  generally  con- 
tinuous, and  passes  first  over  the  larger  wheel,  then  hangs 
in  a  free  loop,  then  passes  over  the  smaller  wheel  and  back 
again  to  the  larger  wheel  through  a  loop.  A  force,  FI,  is 
applied  to  the  chain  just  before  it  passes  over  the  larger 


88 


THEORY  OF  PHYSICS 


[CH.  II 


wheel,  and  this  is  balanced  by  a  force,  t\  applied  in  the 
loop  formed  just  after  the  chain  leaves  the  larger  wheel. 

The  chain  cannot  slip  ;  and  so, 
if  the  forces  are  parallel,  they 
can  be  represented  as  shown  in 
the  diagram.  Let  the  length 
of  one  link,  that  is,  of  one  open- 
ing in  the  groove  of  the  wheel, 
be  #,  and  let  there  be  n  of 
them  in  the  rim  of  the  larger 
wheel  and  m  in  that  of  the 
smaller.  Then  the  radius  of 
the  larger  wheel  is  n  a  /  2  TT  ; 
that  of  the  smaller,  m  a  /  2  TT. 
Hence,  taking  moments  around 
the  axle,  since  there  is  sup- 
posed to  be  equilibrium, 


na       J*9ma       J^na  _ 

1    O  O     O  __       ""     O    O  ' 

Zi  7r          ZiZiTr          £   £  7T 

that  is, 

2F,n-  F2  O  -  m)  =  0, 

«    '^;«SS  ^34> 

As  stated  above,  in  general  m 
is  made  equal  to  n  —  1 ;  hence 
in  this  case 


A     B       0       C     D 


I 

FIG.  65. 


Fa 


So  the  mechanical  advantage  of  a  differential  pulley  is  as 
great  as  desired,  being  conditioned  only  by  the  number  of 
openings  or  teeth  in  the  wheels. 

73.  A  screw  is  simply  a  groove  cut  along  the  line  of  an 
inclined  plane  which  is  wound  around  a  circular  cylinder. 
The  "  pitch  "  of  the  screw  is  the  angle  between  this  plane 


73] 


DYNAMICS 


89 


and  a  plane  perpendicular  to  the  axis  of  the  cylinder,  which 
may  be  called  the  base-plane ;  0  in  the  diagram  is  the  pitch. 
A  length  /  along  the  base-plane  corresponds  to  a  distance 
I  tan  6  parallel  to  the  axis.  If  the  radius  of  the  screw  is  r, 
one  turn  corresponds  to  a  distance  along  the  base  plane,  i.  e. 
around  the  cylinder,  of  2  TT  r,  and  hence  to  a  distance  along 
the  axis  of  2  TT  r  tan  6.  Let  a  moment,  L,  applied  to  the 


9  FIG.  66. 

screw  produce  one  complete  turn,  and  let  it  just  overcome 
a  force,  F,  acting  on  a  nut  which  is  made  to  move  along  the 
screw.  Then,  the  work  done  by  the  moment  is  L  multi- 
plied by  the  angle  (Art.  57),  which  in  this  case  is  2  TT  ;  and 
that  done  against  the  force  F  is  F  2  TT  r  tan  0.  Hence 


And  so 


L  =  Fr  tan  9 


(35) 


Consequently,  I/  tan  0  is  the  mechanical  advantage  of  a 
screw,  and  this  can  be  as  great  a  quantity  as  desired. 

A  differential  screw  can  also  be  made  by  cutting  threads 
of  different  pitches  on  the  inside  and  outside  of  a.  hollow 
circular  cylinder,  and  fitting  a  second  screw  inside  this 
cylinder.  Then,  if  the  outer  screw  is  turned  through  a 


90  THEORY  OF   PHYSICS  [CH.  II 

certain  angle,  the  inner  screw  will  mo-  g  the  axis 

a  distance  which  depends  upon  the  differer.ee  of  the  two 
pitches.     Its  mechanical  advantage  is  \   v\     reat. 

Comparison  of  Translation  and  Rotation. 

T  •  i    distance  an  trie 

Linear  speed,  — ;  angular      eed         F-. 

time  ,ie 

Linear  velocity,   linear  speed  angula:    *     city,  angular  speed 

in  a  particular  direction;  arouim  a  particular  axis. 

Linear  acceleration,  angular  acceleration, 

linear  velocity  angular  velocity 

time  time 

Mass ;  moment  of  inertia. 

Linear  momentum,  mass  X  lin-  angular    momentum,    moment 

ear  velocity ;  of  inertia  X  angular  velocity. 

Force,  mass  X  linear  accelera-  moment,  moment  of  inertia  X 

tion;  angular  acceleration. 

Linear  impulse,  force  X  time;     angular    impulse,    moment   X 

time. 
Work,  force  X  distance;  work,  moment  X  angle. 

Kinetic    energy,     £    mass    X     kinetic  energy,   £  moment  of 
square  of  linear  speed;  inertia  X  square  of  angular 

speed. 

WAVES 

74.  Wave-motion.  By  a  wave  is  meant  the  advance  of  a 
disturbance  into  a  medium ;  the  portions  of  the  medium 
do  not  move  onward  carrying  the  disturbance,  but  they 
hand  it  on  from  one  portion  to  the  next.  Thus  a  stone 
dropped  into  a  basin  of  water  produces  a  wave,  which 
spreads  outward  in  all  directions.  If  a  stretched  cord  has 
one  end  moved  suddenly  sideways,  a  wave  will  pass  along 
the  cord.  If  a  stretched  spiral  spring  has  one  end  moved 
suddenly  in  the  direction  of  the  length  of  the  spring,  a 


74]  DYNAMICS  91 

wave  of  compression  or  rarefaction  will  move  along  the 
spring.  In  all  these  cases  a  certain  state  of  affairs,  a 
certain  condition,  advances  and  forms  the  wave.  The 
individual  particles  of  the  medium  move  only  a  short 
distance.  Tf  stones  are  dropped  into  the  water  at  regu- 
lar intervals,  or  if  the  end  of  the  string  is  moved  side- 
ways and  back  regularly,  or  if  the  end  of  the  spiral  spring 
is  moved  forward  and  back  regularly,  there  will  be  a  suc- 
cession of  waves.  The  waves  on  the  surface  of  the  water 
will  be  ^considered  more  in  detail.  They  consist  of  a 


FIG.  67. 

series  of  crests  and  hollows,  having  at  any  instant,  in  the 
simplest  case,  the  appearance  of  a  sinuous  curve.  The 
advance  of  the  wave  consists  in  the  moving  forward  of 
this  form.  The  distance  from  crest  to  crest  is  called  the 
"  wave-length ; "  the  number  of  crests  which  pass  any 
point  in  one  second  is  called  the  "  wave-number "  or  the 
"  frequency ; "  the  distance  through  which  the  wave  ad- 
vances in  one  second  is  called  its  velocity.  The  symbols 
for  wave-length,  wave-number,  and  velocity  are  X,  n,  u  ; 
and  consequently 

u  =  n\ (36) 

The  individual  particles  of  the  water  are  moving  in  vertical 
planes,  making  harmonic  vibrations.  Adjacent  particles 
are  of  course  in  different  positions  of  their  motions ;  but 
there  is  always  at  a  distance  of  one  wave-length  from  any 
particle  another  particle  whose  motion  is  identical  with  its 


92  THEORY   OF  PHYSICS  [CH.  II 

own,  for  at  an  interval  of  a  wave-length  everything  is 
repeated. 

Similarly,  as  a  wave  advances  along  a  string,  there  are 
crests  and  hollows ;  and  the  definitions  of  wave-length, 
wave-number,  and  velocity  are  the  same  as  above.  The 
individual  particles  of  the  string  move  across  the  length  of 
the  string,  making  harmonic  vibrations.  Such  a  wave  as 
this  is  called  a  "  transverse  "  wave,  because  the  motion  of 
the  particles  is  perpendicular  to  the  direction  of  advance 
of  the  wave. 

In  the  spiral  spring  there  are  condensations  and  rarefac- 
tions at  regular  distances  apart ;  and  the  wave-length  is 
the  distance  from  one  condensation  to  the  next,  etc.  In 
this  wave  the  individual  portions  of  the  spring  are  making 
harmonic  vibrations  backward  and  forward  in  the  line 
of  the  advance  of  the  wave.  Such  waves  are  called  "  longi- 
tudinal." 

If  two  or  more  waves  are  passing  through  a  medium 
at  the  same  time,  the  resultant  effect  at  any  point  of  the 
medium  is  simply  the  geometrical  sum  of  the  separate 
effects  which  each  wave  would  by  itself  have  produced, 
provided  only  that  the  motion  of  the  particles  of  the  me- 
dium is  small  compared  with  the  length  of  the  waves.  Any 
wave,  however  complicated,  may  be  considered  as  composed 
of  a  number  of  the  simplest  waves,  whose  form  is  that 
described  above. 

It  requires  energy  to  produce  waves,  and,  if  a  body 
stops  waves,  it  receives  energy.  So  waves  can  be  regarded 
as  carrying  away  energy  from  a  source,  and  giving  it  to 
whatever  destroys  the  motion.  The  energy  in  a  wave  is, 
naturally,  partly  kinetic  and  partly  potential;  because  the 
portions  of  the  medium  are  in  motion,  and  they  are  also 
displaced  with  reference  to  each  other. 

If  the  wave-length  is  long,  the  waves  will  pass  around 
a  small  obstacle,  as  water-waves  do  around  a  small  pole 
sticking  up  out  of  the  surface :  whereas  a  large  obstacle 


74]  DYNAMICS  93 

will  either  stop  them  or  reflect  them.  If  an  object  stops 
the  waves,  it  is  said  to  "  absorb  "  them  or  to  absorb  their 
energy. 

The  amount  of  energy  carried  by  the  waves  through  an 
area  of  one  square  centimetre  placed  at  any  point  perpen- 
dicular to  the  direction  of  the  waves  is  called  the  "in- 
tensity" of  the  waves  at  that  point.  If  the  waves  are 
sent  out  in  an  isotropic  medium  from  a  point,  they  will 
spread  outward  in  the  form  of  spheres.  If  two  spheres  of 
radii,  r1  and  rz,  be  drawn  around  the  point-source  of  the 
waves,  the  same  amount  of  enefgy  must  be  passing 
through  these  two  spherical  surfaces  at  any  instant,  as  it 
is  carried  outward  by  the  waves.  The  areas  of  the  sur- 
faces of  the  two  spheres  are  4  TT  n2  and  4  TT  r22.  So,  if  E 
is  the  total  amount  of  energy  passing  through  each  of  the 

77F 

surfaces,  /i,  the  intensity  at  the  distance  rz  is  -:  -  -  ;  and 

4  TTTi2 


/2,  the  intensity  at  the  distance  r1}  is  -  -  ^  .     Hence 

47rr22 

/i:7a  =  —  a:i      .....     (37) 
TIZ    rf 

That  is,  the  intensity  of  waves  in  an  isotropic  medium 
varies  inversely  as  the  square  of  the  distance  from  the 
source. 

It  may  also  be  proved  that  the  intensity  of  the  waves 
varies  directly  as  the  square  of  the  amplitude  of  the  vi- 
brations of  the  particles  of  the  medium  which  carries  the 
waves. 

Since  liquids  and  gases  take  the  shape  of  the  vessels 
which  contain  them,  they  do  not  resist  any  attempt  made 
to  displace  one  layer  over  another  ;  and  so  a  wave  like 
a  transverse  wave  in  a  cord  is  impossible  in  them.  The 
ordinary  waves  on  the  surface  of  water  or  of  any  liquid 
are  not  due  to  any  reaction  of  the  liquid  itself,  but  are 
caused  by  the  effect  of  the  earth's  force  of  gravitation 


94  THEORY  OF  PHYSICS  [CH.  II 

(Art.  76).  Liquids  and  gases,  then,  of  themselves  can 
transmit  only  longitudinal  waves  ;  while  solids  can  trans- 
mit both  longitudinal  and  transverse  ones. 

The  velocity  of  any  train  of  waves  in  any  homogeneous 
medium  must  vary  directly  as  the  elasticity  of  the  me- 
dium and  inversely  as  its  inertia.  It  may  be  proved 
mathematically  that  this  is  the  case,  and  that  the  velocity 
depends  upon  nothing  else,  if  there  are  no  obstacles  im- 
mersed in  the  medium,  which  may  affect  waves  differently. 
The  velocity  is  the  same  for  waves  of  all  lengths,  short 
and  long;  and  of  all  ^amplitudes,  provided  only  that  the 
limits  of  elasticity  are  not  reached.  These  facts  are  not 
true  of  waves  on  the  surface  of  water,  because  they  are 
not  due  to  any  elastic  properties  of  the  water;  and  in 
fact  long  water-waves  travel  faster  than  shorter  ones. 
Further,  if  the  medium  is  changed  in  any  way,  as  by 
immersing  a  great  number  of  small  bodies  in  it  which 
affect  waves  differently  according  to  their  wave-numbers, 
the  velocity  of  different  waves  is  no  longer  the  same. 

75.  If  a  wave  of  any  kind  is  sent  out  in  any  medium 
from  a  centre  of  disturbance,  its  progress  is  marked  by  the 
motion  of  the  portions  of  the  medium  when  the  wave 
reaches  them  ;  and  the  locus  of  the  points  which  the  wave 
has  just  reached  is  called  the  "  wave-front."  In  the  case 
of  a  wave  proceeding  out  from  a  point  into  an  isotropic 
medium,  the  wave-front  is  obviously  a  sphere,  and  the 
wave  is  called  a  "  spherical "  one.  If  the  centre  is  very  far 
away,  the  sphere  is  so  large  that  any  small  portion  of  the 
surface  is  practically  a  plane ;  and  so  we  can  speak  of  a 
"  plane  "  wave. 

Consider  a  spherical  wave  whose  centre  of  disturbance 
is  0.  At  a  certain  instant  its  front  passes  through  a 
series  of  points,  PI,  P2,  PB,  etc.,  on  a  sphere  of  radius  R\ 
and  they  are  set  in  vibration.  At  an  interval  of  time 
la,ter  the  wave-front  reaches  another  series  of  points,  Qlt  Qz, 
Q8,  etc.,  on  a  sphere  of  radius  S.  It  is  obvious  from  ordi- 


75] 


DYNAMICS 


95 


nary  geometry  that,  if  spheres  of  the  same  radii,  S—B,  are 
described  around  each  of  the  points  Pi,  P2,  P3,  etc.,  they 
will  just  touch  the  outer  spher- 
ical surface  on  which  lie  the 
points  Qi,  ft,  Q8,  etc.  Conse- 
quently, the  effect  is  just  the 
same  at  the  outer  spherical  sur- 
face if  the  spherical  wave  itself 
advances  there,  or  if  the  origi- 
nal wave  be  considered  replaced 
at  any  instant  by  a  great  num- 
ber of  smaller  spherical  waves 
sent  out  by  each  of  the  points 
Pi,  Pz,  P3,  etc.,  of  the  original 
wave-front  at  that  instant ;  be- 
cause the  time  taken  for  the 
original  wave  to  pass  from  the 
first  position  to  the  second  is 

Sf       7? 

— ,  where  u  is  the  velocity  of  the  wave ;  and  in  this 

iv 

same  time  the  smaller  spherical  waves  from  Pi,  P2,  Ps,  etc., 
will  just  reach  the  outer  surface.  This  process  is  said  to 
be  "  breaking  up  "  a  "  primary  "  wave  into  "  secondary  " 
waves.  Although  the  effect  at  the  wave-front  is  the  same, 
however  it  ie  described,  in  terms  of  primary  or  secondary 
waves,  it  is  often  convenient  from  a  mathematical  stand- 
point to  refer  the  disturbance  to  the  secondary  waves 
rather  than  to  the  primary  one. 


CHAPTEK   III 
GRAVITATION 

76.  Weight.  Keference  has  been  made  several  times  to 
the  fact  that  all  bodies  have  their  motion  influenced  by 
the  presence  of  the  earth  ;  and  it  was  stated  that  any 
body  falling  toward  the  earth  has  a  constant  acceleration 
which  is  the  same  for  all  bodies  at  any  one  place  on  the 
earth.  This  constant  acceleration  was  called  g\  and  its 
numerical  value  is  nearly  980  in  the  C.  G-.  S.  system. 
This  fact,  that  all  bodies,  whatever  their  material  or  mass, 
fall  towards  the  earth  with  the  same  acceleration,  is  most 
remarkable.  It  may,  however,  be  proved  by  experiment  in 
many  ways.  1.  Allow  bodies  of  various  kinds  and  masses 
to  fall  inside  a  vacuum.  All  resistance  of  the  air  is  now 
removed,  and  the  bodies  are  observed  to  fall  side  by  side. 
2.  It  was  proved  in  Article  51  that  the  period  of  a  pendu- 
lum was  T  =  2  TT  y  -  ;  and  so  g  —  /yT2  ;  I  and  T  can  both 

J 

be  accurately  measured ;  and  thus  g  may  be  determined . 
It  is  found  that  the  same  value  of  g  is  thus  obtained  when 
the  pendulum  is  made  of  any  body  or  has  any  mass. 
Consequently  the  force  "  acting  on  "  any  body  of  mass,  m, 
towards  the  earth  is  m  g ;  and  this  force  is  often  called  the" 
"  weight "  of  the  body  with  reference  to  the  earth. 

Since  a  chemical  balance  can  measure  forces,  it  can  com- 
pare the  weights  of  two  bodies.  Place  one  body  in  each 
pan  of  the  balance,  and  alter  the  mass  of  one  by  adding  to 
it  or  taking  away,  until  the  two  weights  are  equal.  Let 


77]  GRAVITATION  97 

the  masses,  as  measured  by  inertia,  be  m  and  mi.  Then, 
as  their  weights  are  equal, 

m  g  =  ml  g, 

* 

and  consequently  m  =  ml}  since  at  the  same  place  g  is  the 
same  for  all  bodies  ;  and  so  the  masses  as  measured  by 
inertia  are  also  equal.  This  is  equivalent  to  saying  that, 
if  two  bodies  have  the  same  mass  as  measured  by  inertia, 
they  also  have  the  same  weight.  Consequently  equal 
masses  might  perfectly  well  be  defined  as  corresponding  to 
equal  weights  instead  of  as  corresponding  to  equal  inertias  : 
the  two  definitions  are  identical  since  g  is  proved  by  ex- 
periment to  be  a  constant,  the  same  for  all  portions  of 
matter. 

For  different  points  of  the  earth's  surface  g  is  different ; 
and  so  the  weight  of  a  body  varies,  although  its  mass  does 
not.  But,  if  two  bodies  have  the  same  weight  at  any  one 
place,  they  will  also  agree  at  any  other  place.  The  chief 
causes  of  the  variations  in  g  are  the  rotation  of  the  earth, 
the  ellipsoidal  shape  of  the  earth,  and  local  disturbances 
such  as  mountains  and  differences  in  height  above  sea- 
level. 

77.  Universal  Gravitation.  This  effect  of  the  earth  on 
a  body  near  it  is  only  a  special  case  of  a  more  general 
law  of  nature.  There  seems  to  be  undoubted  evidence 
that  each  portion  of  matter  in  the  universe  influences  the 
motion  of  every  other  portion  ;  and  a  law  has  been  pro- 
posed which  seems  to  be  in  accord  with  all  the  observed 
facts,  except  possibly  when  the  portions  of  matter  are  as 
small  and  as  near  together  as  are  the  molecules  of  a  body. 
This  law,  known  as  Newton's  Law  of  Universal  Gravi- 
tation is,  that,  if  two  bodies  whose  masses  are  mi  and  ra2 
are  at  a  distance,  r,  apart,  where  r  is  large  compared  with 
the  size  of  the  bodies,  their  mutual  action  is  such  that,  if 
free  to  move,  they  approach  each  other  with  a  force  which 


98  THEORY  OF  PHYSICS  [CH.  Ill 

is  proportional  to  the  product  of   their  masses   and   in- 
versely to  the  square  of  their  distance  apart,  i.  e.   , 


(1) 


where  7  is  simply  a  constant  of  proportionality. 

It  may  be  proved  without  much  difficulty  that,  if  either 
of  the  bodies  is  a  homogeneous  sphere,  its  action  is  exactly 
what  would  be  produced  if  the  entire  mass  was  concen- 
trated at  its  centre. 

Ordinary  weight  with  reference  to  the  earth  is  an  illus- 
tration of  the  law  ;  because,  if  mi  is  the  mass  of  the  body, 
mz  that  of  the  earth,  and  r  the  distance  from  mi  to  the 
centre  of  the  earth,  7  m2  /  rz  is  a  constant  entirely  inde- 
pendent of  mi.  Hence  F  =  m\g.  Further,  the  forces  due 
to  the  earth  "  acting  on  "  different  portions  of  matter  near 
each  other  are  all  parallel,  because  the  centre  of  the  earth 
is  so  far  away  from  the  surface  that  lines  drawn  to  it  from 
bodies  near  each  other  on  the  surface  are  parallel. 

The  law  has  been  directly  verified  in  many  ways  :  (1)  by 
direct  measurement  of  the  forces  between  different  masses 
at  varying  distances  ;  (2)  by  the  application  of  deduc- 
tions from  it  to  astronomical  calculations  and  predictions. 
An  application  of  this  last  method  is  to  predict  the  period 
of  the  revolution  of  the  moon  around  the  earth.  In  Article 
19  it  was  proved  (Formula  8)  that  the  period  of  a  body 
moving  in  a  circle  of  radius  r,  when  the  acceleration  to- 
wards the  centre  is  a,  is 

T  =  2  TT  V'rfa. 

Call  the  mass  of  the  earth  w2,  that  of  the  moon  mi.     Then 
the  acceleration  at  the  moon  due  to  the  earth  is 

F          m, 
a  —  —  =  7  -  „  , 
ml  r2 

where  r  is  the  radius  of  the  moon's  orbit. 


78]  GRAVITATION  99 

Similarly,  the  acceleration  at  the  surface  of  the  earth  is 

-—^ ,  where  r\  is  the  radius  of  the  earth.     But  r,  the  dis- 
r,* 

tance  from  the  centre  of  the  earth  to  the  moon,  is  very 
nearly  60  r\ ;  and  the  acceleration  at  the  surface  of  the 
earth  is  known  to  be  nearly  980.  Hence 


Eliminating  7, 


Therefore, 


=  2*V; 


60  x  4  X  109 

r  =  60  TI  =  -  - 


and  so  T  =  27  days  6  hours,  whereas  the  observed  period 
is  27  days  8  hours,  which  is  very  close  agreement. 

The  formation  of  the  tides  in  the  oceans  is  also  an  illus- 
tration of  the  influence  of  matter  upon  matter  ;  in  t  is 
case,  of  the  sun  and  moon  upon  the  fluid  portions  of  je 
earth  ;  and  the  main  features  of  the  tides  may  be  et-  dly 
predicted  by  elementary  applications  of  the  general  law  of 
gravitation.  There  are,  of  course,  innumerable  other  illus- 
trations of  the  truth  of  the  law. 

78.  Centre  of  Gravity.  There  is,  then,  a  force  due  to  the 
earth  "  acting  on  "  every  portion  of  matter  ;  and,  if  the 
matter  is  near  the  earth,  this  force  has  for  its  value  m  g, 
and  the  forces  acting  on  the  different  portions  of  matter  of 
the  same  system  are  parallel.  As  was  proved  in  Article 
55,  if  parallel  forces  act  on  a  system  of  bodies,  each  force 
having  for  its  value  mg,  the  resultant  force  has  for  its 
value  Mg  where  M  is  the  sum  of  the  separate  masses,  and 
its  line  of  action  is  parallel  to  the  component  forces  and 
passes  through  the  centre  of  inertia.  A  special  case  of  a 


100 


THEORY  OF  PHYSICS 


[CH.  Ill 


system  of  particles  is  a  single  large  body ;  and  so  in  all 
cases  there  is  a  certain  point,  viz.  the  centre  of  inertia,  at 
which  the  whole  resultant  force  may  be  supposed  to  act. 
For  this  reason  the  centre  of  inertia  of  any  body  or  system 
is  often  called  its  "  centre  of  gravity ;  "  and  its  position 
may  be  calculated  as  was  shown  in  Article  40. 

In  any  actual  case  it  may  be  found  by  experiment. 
Suspend  the  body  so  that  it  is  free  to  turn  around  a  hori- 
zontal axis  at  any  point,  P ;  and  let  C 
be  the  centre  of  inertia.  There  is  a 
force,  mg,  vertically  downward  through 
C]  and  this  force  will  have  a  moment 
around  the  axis  at  P  unless  C  is  in  a 
vertical  line  through  P.  Consequently, 
if  C  is  not  in  that  line,  the  body  will 
turn  around  the  axis  until  it  does  come 
there.  (In  general  the  body  will  make 
oscillations  about  the  axis,  and  such  a 
vibrating  body  is  called  a  "  compound  " 
pendulum,  to  distinguish  it  from  a  sim- 
ple pendulum.  See  Article  51.  If  there 
is  friction,  the  body  will  soon  come  to 
rest.)  So  it  is  known  that,  if  the  body  is  at  rest,  C  must 
be  somewhere  in  a  vertical  line  passing  through  P.  Now 
suspend  the  body  so  that  it  is  free  to  turn  around  another 
horizontal  axis  not  in  this  line ;  and,  when  it  comes  to 
rest,  the  centre  of  inertia  must  be  somewhere  in  a  vertical 
line  passing  through  the  new  point  of  support.  Since,  then, 
the  centre  of  inertia  must  lie  on  both  these  lines,  it  must 
be  the  point  of  their  intersection. 

If  a  body  is  so  balanced  that  when  it  is  at  rest  the 
centre  of  gravity  lies  vertically  above  the  axis,  it  is  in 
unstable  equilibrium ;  because,  if  it  is  displaced  at  -all,  it 
will  immediately  turn  over  so  that  its  centre  of  gravity 
may  get  as  near  the  earth  as  possible.  It  will  come  to 
rest  again  (if  there  is  friction)  with  its  centre  vertically 


FIG 


78]  GRAVITATION  101 

below  the  axis,  where  the  equilibrium  is  stable.  This  is 
an  illustration  of  the  general  principle  that  the  potential 
energy  of  a  system  always  tends  to  become  as  small  as 
possible.  If  the  axis  passes  through  the  centre'  of  inertia, 
the  body  is  in  neutral  equilibrium ;  for  any''  displacement, 
produces  a  constant  permanent  effect"  Li  general'/ fchafe- 
fore,  a  body  is  in  stable  equilibrium  with  reference  to  tfre 
earth,  if  its  centre  of  gravity  is  as  near  the  earth  as  it  can 
get  under  the  existing  conditions  of  restraint. 

Units 

Length  :    centimetre  (cm.). 

1  cm.  =  0.39370  inches  =  0.032809  feet. 
1  in.   =  2.5400  cm. 

Mass  :    gram  (g.). 

1  g.   =  0.0022046  pounds. 
1  Ib.  =  453.59  g. 

Time  :    mean  solar  second  (sec.). 

Speed  :    1  cm.  per  sec. 

Acceleration  :    (1  cm.  per  sec.)  per  sec. 

Force :   dyne.     1  g.  given  unit  acceleration  in  1  sec. 

Mass  m.  has  weight  mg.  dynes. 

1  gram  weighs  g.  dynes;  1  pound  weighs  4.45  x  105  dynes. 

Energy  :    erg.     1  dyne  X  1  centimetre. 

1  foot-pound  =  1.3562  x  107  ergs. 

Activity  :    Watt.     107  ergs,  per  sec. 

1  horse-power  =  7.46  X  102  Watts. 

Acceleration  due  to  the  earth. 

Boston,  980.4.  Denver,  979.6. 

Philadelphia,  980.2.  San  Francisco,  979.95. 

Washington,    980.1.  Greenwich,  981.17. 

Chicago,  980.3.  Paris,  980.96. 

St.  Louis,         979.99.  Berlin,  981.24. 


V      CHAPTEE    IV 

PROPERTIES  OF   SIZE  AND  SHAPE  OF  MATTER 

79.  Introduction.  Both  these  general  properties  of  mat- 
ter, inertia  and  weight,  are  common  to  all  forms  of  matter, 
solids,  liquids,  and  gases  ;  and  all  the  laws  which  have 
been  stated  apply  equally  well  to  them  all.  There  are, 
however,  certain  other  properties  which  are  different  for 
various  forms  of  matter.  As  noted  in  Article  3,  the  name 
"  solid  "  is  commonly  given  to  a  body  which  has  a  volume 
and  a  shape  of  its  own ;  the  name  "  liquid,"  to  a  body 
which  has  a  volume  of  its  own,  but  which  assumes  the 
shape  of  the  containing  vessel ;  the  name  "  gas,"  to  a  body 
which  has  neither  volume  nor  shape  of  its  own,  but  as- 
sumes those  of  the  containing  vessel.  These  definitions 
are  by  no  means  perfectly  exact ;  but  they  satisfy  all  gen- 
eral purposes. 

The  volume  and  shape  of  any  body  may  be  changed ; 
and  the  behavior  of  the  body  after  these  changes  are  pro- 
duced distinguishes  definitely  between  different  forms  of 
matter.  If  either  the  shape  or  the  volume  of  a  solid  is 
altered,  there  is  in  general  a  tendency  for  the  body  to 
return  to  its  previous  condition ;  and  so  work  is  required 
to  produce  the  change,  and,  if  the  cause  producing  the 
change  ceases  to  act,  the  body  will  return  to  its  previous 
state.  Various  solids  differ  widely  as  to  these  properties. 
If  the  volume  is  changed  with  ease,  the  solid  is  said  to  be 
"  compressible ; "  while,  if  the  shape  is  changed  with  diffi- 
culty, the  body  is  said  to  be  "  rigid."  Thus  a  piece  of 
steel  is  not  easily  compressible  and  is  rigid ;  a  piece  of 


79]      PROPERTIES  OF  SIZE  AND  SHAPE  OF  MATTER       103 

cork  is  very  rigid ;  a  piece  of  rubber  is  compressible  and 
not  rigid.  In  all  these  cases,  though,  the  body,  if  not 
changed  too  much,  will  return  to  its  previous  state  if 
allowed  to ;  and  such  a  body  is  said  to  be  "  elastic."  In 
some  solids,  however,  when  a  change  is  produced,  it  be- 
comes partially  or  completely  permanent;  thus,  after  a 
piece  of  lead  or  putty  is  changed  in  shape  or  volume,  it 
does  not  completely  recover  its  former  condition.  The 
molecules  of  the  body  have  been  permanently  displaced 
over  each  other ;  and  part  of  the  work  done  has  been  pro- 
duced by  the  passage  of  energy  into  the  molecules,  and 
so  a  "  heat-effect "  is  produced.  Such  bodies  are  called 
"  plastic  "  or  "  inelastic." 

A  perfect  liquid  would  offer  no  resistance  at  all  to  a 
change  in  shape ;  that  is,  one  layer  could  move  over  an- 
other with  perfect  freedom.  But  no  actual  liquid  has  this 
property ;  and  so,  when  one  portion  of  a  liquid  moves  over 
another  there  is  always  a  resistance  which  tends  to  make 
the  motion  of  the  two  portions  the  same.  This  property 
is  called  "viscosity."  Any  force,  though,  however  small, 
will  make  one  portion  of  the  liquid  move  over  another,  if 
only  it  is  applied  long  enough.  All  liquids  are  more  or 
less  compressible  ;  and  all  tend  to  recover  their  former 
volume  when  the  force  is  removed ;  that  is,  all  liquids  are 
elastic.  ^ 

A  perfect  ga?  would  also  not  be  viscous ;  but  all  actual 
gases  are  more  or  less  so.  All  gases  are  compressible  ;  and 
they  offer  a  resistance  to  a  decrease  in  volume.  If  the  vol- 
ume is  restored  to  its  original  value,  the  gas  again  ex- 
pands ;  and  so  it  may  be  said  to  be  elastic. 

Some  solids  and  all  liquids  and  gases  are,  then,  elastic, 
if  the  change  produced  is  not  too  great ;  and  this  limiting 
change  beyond  which  a  body  cannot  go  without  losing  the 
power  to  recover  its  previous  condition  is  called  the  "  limit 
of  elasticity."  The  change  is  always  produced  by  some 
force ;  arid  the  amount  of  the  change  depends  upon  the 


104  THEORY  OF  PHYSICS  [CH.  IV 

amount  of  the  force,  the  area  over  which  it  acts,  the  amount 
of  the  property  which  is  changed,  and  the  nature  of  the 
body  itself.  To  simplify  this  statement,  various  definitions 
i  have  been  adopted.  The  internal  "  stress  "  is  the  force  of 
reaction  produced  in  the  body  by  the  applied  force,  divided 
by  the  area  over  which  the  reaction  acts.  The  "  strain  "  is 
the  amount  of  the  change,  divided  by  the  amount  of  the 
quantity  changed.  The  "  coefficient  of  elasticity "  for 
any  particular  kind  of  change  in  any  body  is  the  ratio 
of  the  stress  to  the  strain.  And  experiments  prove  that 
for  all  strains  within  the  limits  of  elasticity,  the  coefficient 
of  elasticity  corresponding  to  any  particular  type  of  change 
is  a  constant  for  any  one  substance.  This  is  called 
"  Hooke's  Law."  It  is  equivalent  to  saying  that  the  strain 
is  proportional  to  the  stress. 

•s 

SPECIAL  PROPERTIES  OF  SOLIDS 

80.  A  solid  has  both  a  volume  and  a  shape  of  its  own ; 
and  therefore  any  elastic  solid  has  two  independent  co- 
efficients of  elasticity,  —  one  corresponding  to  a  change  in 
volume,  the  other  to  a  change  in  shape. 

81.  Change  in  Volume.     To  change  the  volume  of  a  solid 
without  changing  its  shape  or  those  of  its  smallest  por- 
tions is  not  easy ;  for  it  is  necessary  to  apply  a  uniform 
stress  all  over  the  body  perpendicular  to  its*surface  at  each 
point.     It  can  be  done,  however,  by  immersing  the  solid  in 
a  liquid  and  then  increasing  the  pressure  of  the  liquid.    In 
defining  the  coefficient  of  elasticity  it  was  necessary  to 
specify  the  internal  stress,  because  inelastic  bodies  have 
no  internal   reaction  when  a  force  is  applied  to  change 
them.     But,  if  a  body  is  elastic,  there  is  a  reaction  ;  and 
the   internal    force   equals    the    external    force   when    a 
change  is  produced.     Let  the  stress  producing  the  change 
in  volume  be  p  where  p  is  the  force  divided  by  the  area ; 
and   let   the  original  volume  be  v,  and  the  change  pro- 


82]        PROPERTIES  OF  SIZE  AND  SHAPE  OF  MATTER       105 


duced  by  p  be  A  v.     The  strain  is  then  A  v  /  v,  and  the 
coefficient  of  elasticity  for  a  change  in  volume  is,  therefore, 


k 


A  v 


This  coefficient  is  sometimes  called  the  "  bulk-modulus  ; " 
and  it  has,  of  course,  different  values  for  different  solids. 
Some  of  the  values  are,  — 

For  steel,  k  =•  18.4    X  1011 ; 

"     flint  glass,    k=    4.15  X  10n  ; 
"     cast-iron,      k  =    9.64  X  1011. 


7 


B 

i  1  

i 
i 
i 

_i_ 

C 

1 

/ 

/ 
/ 
/ 

I    I 

82.  Change  in  Shape.  To  change  the  shape  of  a  solid 
without  altering  its  volume  sensibly  is  not  difficult.  Nail 
firmly  two  boards  to 
opposite  ends  of  a  block 
of  wood ;  then,  holding- 
one  board  firmly  fixed, 
push  the  other  side- 
ways in  the  direction 
of  its  own  plane.  The 
shape  will  be  changed, 
as  shown  ;  the  volume, 
however,  will  remain 
sensibly  the  same.  If 
the  force  is  F,  and  the 
area  of  the  top  of  the 
block  A,  the  stress  is 
F/A.  Call  this  T.  A 
measure  of  the  strain 
is  the  angle,  0,  between 
the  former  direction 

of  the  edge  of  the  block  and  the  new  one  produced  by  the 
stress.  The  coefficient  of  elasticity  for  a  change  in  shape 
is,  then. 

»=/* (2) 


BB 


D 
FIG.  70. 


106  THEORY  OF  PHYSICS  [CH.  IV 

This   coefficient   is    sometimes    called   the    "coefficient  of 
rigidity." 

A  change  in  shape  is  also  produced,  without  a  change  in 
volume,  when  a  rod  or  wire  is  twisted  around  its  axis  of 
figure.  This  is  easily  seen  if  the  alteration  in  any  small 
cubical  portion  of  the  rod  or  wire  is  considered.  If  one 
end  of  the  rod  or  wire  is  held  firm,  and  the  other  twisted 
by  a  moment  L  around  the  axis  of  figure,  the  angle  0 
through  which  the  lower  end  will  be  turned  is  given  by 
the  following  formula,  which  may  be  deduced  by  the  help 
of  higher  mathematics  :  — 


where  I  is  the  length  of  the  rod  or  wire  and  B  is  a  con- 
stant for  any  one  rod  or  wire,  depending  upon  its  dimen- 
sions. For  a  circular  cylinder  of  radius  r,  B^  =  TT  r*  /  2  ; 
and  so,  for  a  wire  of  circular  cross-section, 


TT  n  ') 


If  such  a  wire  be  twisted  and  then  set  free  and  allowed  to 
make  torsional  vibrations,  the  moment  tending  to  oppose 
the  motion  at  any  instant,  due  to  the  reaction  of  the  wires 
will  be 


Consequently,  if  a  disc  whose  moment  of  inertia  around 
the  axis  is  /  is  fastened  to  the  free  end  of  the  wire,  its 
angular  acceleration  at  any  instant  of  the  vibration  will  be 

Z//or  #;  and  so  the  vibrations  will  be  harmonic 

(see  Arts.  25  and  51),  and  the  period  of  one  complete 
vibration  will  be 

(5) 


83]       PROPERTIES  OF  SIZE  AND  SHAPE  OF  MATTER       107 

The  quantities  T,  I,  I,  r,  can  be  easily  measured ;  and  so  n, 
the  coefficient  of  rigidity,  may  be  determined  for  the  sub- 
stance of  which  the  wire  is  made.  It  has  been  found  that 

For  steel,  n  =  8.2  X  1011 ; 
"  glass,  n  =  2.4  X  1011 ; 
"  cast-iron,  n  =  5.3  X  1011. 

The  ordinary  vibrations  of  a  spiral  spring  are  due  to  the 
fact  that,  when  the  spring  is  elongated,  the  wire  itself  is 
twisted  ;  and  so  the  vibrations  are  really  torsional  ones. 

83.  In  general,  unless  special  precautions  are  taken,  there 
will  be  a  change  in  both  volume  and  shape ;  and  one  of 
the  most  common  alterations  is  that 
when  a  rod  or  wire  is  compressed 
or  stretched  by  forces  applied,  par- 
allel to  the  axis  of  figure,  at  the 
two  ends.  By  considering  any 
small  cubical  portion  of  the  rod  or 
wire,  it  will  be  obvious  that  its 
shape  is  changed;  and  the  volume 
will  also  be  in  general.  Thus,  let 
a  wire  having  a  circular  cross-sec- 
tion of  radius  r,  and  a  length  /,  be 
fastened  firmly  at  one  end  and  be 
stretched  by  a  force  F  applied  at 
the  lower  end.  Let  the  change  in 
length  produced  by  F  be  A  /.  Then 
the  stress  p  =  F /  TT  r2 ;  the  strain  is  FlG  71 

Al/l',   and  the  coefficient  of  elas- 
ticity, which  in  this  case  is  called  "  Young's  modulus,"  is 


TT  r2  .  A  I 


(6) 


These  quantities  can  all  be  easily  measured ;  and  so  E  can 
be  determined. 


108 


THEORY  OF  PHYSICS 


[CH.  IV 


For  steel,  E  =  2l.  x  1011 ; 
"  glass,  E  =  6.  x  1011 ; 
"  cast-iron,  E  =  13.5  x  1011. 

Since  E  is  a  constant  for  any  one  substance  for  slight 
changes,  A  I  is  proportional  to  F ;  that  is,  the  change  in 
length  is  proportional  to  the  stretching  force.  The  same 
statements  and  formula  apply  also  to  the  case  of  a  rod 
being  compressed  by  forces  applied  at  the  two  ends,  e.  g. 
a  pillar  supporting  a  building. 

Since,  in  determining  Young's  modulus,  there  is  a  change 
in  both  shape  and  volume,  E  must  depend  upon  k  and  n ; 
and  it  may  be  proved  from  theoretical  considerations  that 


L-    *    4.J- 

E      9  k      3  n 


(7) 


Consequently,  if  E  and  n  are  determined  for  any  one  sub- 
stance, k  may  be  calculated ;  and  in  fact  this  is  the  way 
its  value  is  nearly  always  found.  (The  nature  of  a  sub- 


85]        PROPERTIES  OF  SIZE  AND  SHAPE  OF  MATTER       109 

stance  is  often  changed  when  it  is  drawn  out  into  a  wire  ; 
so  the  calculated  value  of  k  applies  to  the  substance  in  the 
form  of  a  wire.)  Another  illustration  of  a  change  in  both 
volume  and  shape  is  when  a  rod  is  bent.  If  a  rod  of 
breadth  b  and  depth  d  is  placed  upon  two  knife  edges  at  a 
distance  I  apart,  and  if  a  force,  Ft  parallel  to  the  depth  of 
the  rod,  is  applied  at  a  point  half-way  between  the  knife 
edges,  the  displacement  of  that  point  in  the  direction  of  the 
force  may  be  proved  to  be 


A 
A  = 


E  is  Young's  modulus  ;  and  so  it  can  be  determined  by 
this  experiment  as  well  as  by  the  one  described  above. 

SPECIAL  PROPERTIES  OF  LIQUIDS 

84.  Introduction,     The  distinguishing  characteristics  of  a 
liquid  are  :  1.  Any  force,  no  matter  how  small,  can  cause 
one  portion  to  move  over  another.     2.  It  is  only  slightly 
compressible,  and  so  keeps  a  definite  volume.     As  a  conse- 
quence, a  liquid  may  be  considered  as  occupying  a  certain 
volume  and  so  having  a  free  surface.     3.  Its  smallest  por- 
tions or  molecules  are  in  motion  in  all  directions  within  the 
volume,  on  the  kinetic  theory  of  matter. 

These  ideas  of  the  properties  of  a  liquid  are  derived,  of 
course,  from  observation  and  experiment  ;  and  every  deduc- 
tion from  these  properties  is  found  to  be  true.  A  few  of 
the  most  important  of  these  deductions  are  the  following. 

85.  Properties  of  a  Liquid  at  Rest.     1.  The  thrust  or  force 
against  any  surface  in  contact  with  a  liquid  which  is  at 
rest  (that  is,  not  flowing)  is  always  perpendicular  to  it. 
For,  if  the  thrust  were  not  perpendicular,  it  would  have  a 
component  parallel  to  the  surface;    and  this  component 
would  produce  flowing  of  the  liquid,  which  is  contrary  to 
the  hypothesis  that  it  is  at  rest.      (See  Fig.  73.) 


110 


THEORY   OF  PHYSICS 


[CH.  IV 


86.  2.  At  any  point  in  a  liquid  at  rest  the  pressure  is  the 
same  in  all  directions.  By  "  pressure  "  is  meant  the  force 
acting  over  a  surface  divided  by  the  area  of  the  surface ;  i.  e. 
if  it  is  uniform,  it  is  the  force  per  square  centimetre.  So 
the  pressure  in  any  direction  at  a  point  is  the  force  acting 
on  an  infinitesimal  surface  at  the  point,  perpendicular  to 
that  direction,  divided  by  the  area  of  the  surface  as  it  is 
made  smaller  and  smaller. 


FIG.  73. 


FIG.  74. 


Consider  the  forces  acting  on  any  small  portion  of  the 
liquid  enclosed  in  a  volume  whose  cross-section  is  a  tri- 
angle A  a  b  c,  and  whose  height  is  any  small  distance.  Call 
the  three  forces  perpendicular  to  the  three  faces  FI,  Fz,  Fs, 
and  the  areas  of  the  faces  Aly  A2,  As.  Then  the  three  pres- 
sures, Pi,p2,ps,  are  FI  /  Ai,  F2  /  A2,  F3  /  As.  If  the  volume 
is  made  small  enough,  these  three  forces  may  be  considered 
as  acting  at  a  point ;  and,  since  by  hypothesis  the  liquid  is 
not  flowing,  the  sum  of  the  components  of  the  forces  in  any 
direction  must  equal  zero.  Take  this  direction  parallel  to 
the  face  on  which  F&  is  acting.  The  component  of  Fl  is 
FI  sin  (u  b  c)  ;  that  of  F2  is  —  F2  sin  (jb  a  c)  ;  that  of  F3  is  0. 


Hence 


F1  sin  (ale)  —  F2 sin  (lac)  =  0. 


Hence,  substituting  the  values  of  the  two  corresponding 
pressures, 

PI  AI  sin  (ale)  —  pz  Az  sin  (I  a  c). 


88]       PROPERTIES  OF  SIZE  AND  SHAPE  OF  MATTER       111 

But  by  geometry,     AI  sin  (a  b  c)  —  A2  sin  (I  a  c), 
and  so  PI  =  pz. 

Since  the  triangle  taken  is  any  triangle,  the  two  pressures, 
pl  and  p2,  are  pressures  in  any  two  directions;  conse- 
quently the  theorem  is  deduced.  (This  theorem  may  also 
be  proved  to  be  true  of  a  liquid  which  is  flowing.) 

87.  3.  The  free  surface  of  a  liquid  at  rest  is  perpen- 
dicular to  the  forces  acting.     Thus  any  surface  of  liquid, 
not  too  small,  on  the  earth,  is  perpen- 
dicular to  the  force  of  gravity ;  for,  if 

it  were  not  so,  the  force  would  have  a 
component  parallel  to  the  surface,  and 
there  would  be  a  flow  of  the  liquid. 
(Where  a  liquid  meets  a  solid,  the  sur- 
face is  not  horizontal  but  curved,  for  FIG.  75. 
reasons  to  be  explained  later.) 

88.  4.  The  pressure  at  any  point  in  a  liquid  at  rest  is 
due  to  two  causes :  (a)  the  containing  vessel,  (b)  the  fact 
that  layers  near  the  bottom  of  a  liquid  must  support  the 
weight  of  the  liquid  above  them. 

Consider  these  two  causes  separately.  To  do  away  with 
the  second,  the  experiments  may  be  considered  as  being 
performed  at  the  centre  of  the  earth.  If  the  liquid  is 
enclosed  in  a  vessel  which  it  completely  fills,  the  pressure 
must  be  the  same  at  all  points  throughout  the  liquid  as  at 
all  points  against  the  walls  of  the  vessel.  This  is  a  direct 
consequence  of  the  fact  that  the  molecules  of  the  liquid 
are  moving  freely  about  in  all  directions ;  and  so,  if  the 
pressure  were  greater  at  one  point  than  another,  the  liquid 
would  immediately  flow  so  as  to  make  the  pressures  the 
same. 

An  illustration  of  this  fact  is  the  "  hydraulic  press," 
which  in  principle  consists  of  two  cylinders  of  different 
radii,  closed  by  pistons  and  connected  by  a  tube.  Let  the 
entire  space  be  filled  with  a  liquid,  and  call  the  areas  of 


112 


THEORY  OF  PHYSICS 


[CH.  IV 


the  two  pistons,  Al  and  A2.     If  a  force,  Fit  acts  on  Alt  the 
force  F2  which  must  be  applied  to  A2  in  order  to  balance 


F2     A,  = 


Aly 


since  the  pressures  must  be  the  same.  Consequently  a 
small  force  acting  over  a  small  area  may  produce  as  great 
a  force  as  is  desired  simply  by  increasing  the  area  on  which 
it  acts.  This  is  not  contrary  to  the  conservation  of  energy, 
because,  if  the  smaller  force  produces  any  motion,  the  lar- 
ger force  will  be  overcome  through  a  distance  smaller  in 
proportion. 


A2 


FIG.  76. 


FIG.  77. 


In  the  case  of  a  vessel  of  water  with  a  free  surface,  such 
as  is  ordinarily  dealt  with,  the  weight  of  the  air  on  the  sur- 
face produces  a  uniform  pressure  throughout  the  liquid  just 
as  if  the  vessel  was  closed  by  a  piston  which  was  pressed 
down  by  a  force  equal  to  the  weight  of  the  atmosphere  on 
the  surface.  (If  the  surface  is  very  small,  there  is  an  ad- 
ditional pressure  called  the  capillary  pressure,  which  will 
be  discussed  later  on.  See  Article  96.) 

89.  Pressure  due  to  Weight.  In  addition  to  this  pressure 
due  to  the  surface,  there  is  the  additional  one,  at  any  point 
in  a  liquid,  due  to  the  weight  of  the  liquid  above  it.  If 
the  liquid  is  not  flowing,  consider  the  forces  acting  on  any 
surface  whose  area  is  A,  which  is  parallel  to  the  free  sur- 


89]        PROPERTIES  OF  SIZE  AND  SHAPE  OF  MATTER     113 


face  but  at  a  depth  h  below  it.  The  force  down  on  this 
surface  is  equal  to  the  weight  of  the  atmosphere  on  an  area 
A  of  the  upper  surface,  vertically  over 
the  lower  one,  plus  the  weight  of  the 
liquid  above  it.  The  volume  of  this  ver- 
'tical  column  of  liquid  is  h  A  ;  hence,  if 
p  is  the  density  of  the  liquid,  the  mass 
is  phA,  because  by  definition  the  den- 
sity is  the  mass  divided  by  the  volume  ; 
and  so  the  weight  is  p  g  h  A.  Conse- 
quently, the  force  downward,  due  to  the 
weight  of  the  liquid,  is  pghA;  and  the 
pressure  downward  therefore  is  p  g  h.  But,  as  stated  above, 
the  pressure  at  any  point  of  the  liquid  is  the  same  in  all 
directions.  Hence  the  pressure  at  any  point,  due  to  the 
liquid  itself,  is 

(9) 


FIG.  78. 


The  entire  pressure  at  any  point  at  a  depth  h  below  the 

surface  is,  then,  the  sum  of  the  pressure  on  the  surface 

and  pgh. 

It  is  seen  that  this  last  pressure,  pg  h,  does  not  depend  in 

the  least  on  the  shape  of  the  containing  vessel,  but  only 

on  the  depth  below  the  free  surface. 

If  a  vessel  having  the  shape  of  a 
frustum  of  a  cone  with  its  smaller  end 
closed,  is  filled  with  a  liquid  to  the 
height  h,  the  pressure  on  the  bottom, 
due  to  the  weight  of  the  liquid,  is  p  g  h. 
Hence,  if  the  area  of  the  bottom  is  A, 
the  entire,  force  on  the  bottom,  due  to 
the  liquid  itself,  is  A  pgh,  which  is  the 
FIG  79  weight  of  a  cylindrical  column  of  the 

liquid  of  cross-section  A  and  height  h. 

The  weight   of  the  rest   of   the  liquid   is   borne  by  the 

walls. 


114 


THEORY   OF  PHYSICS 


[CH.  IV 


FIG.  80. 


If  the  vessel  which  contains  the  liquid  has  the  shape  of 
a  frustum  of  a  cone  with  its  larger  end  closed,  the  pres- 
sure on  the  bottom,  due  to  the  liquid,  is 
pgh;  and  if  the  area  of  the  bottom  is  At 
the  entire  force  produced  on  it  is  therefore 
A  p  g  h.  The  weight  of  the  entire  liquid  is 
not  this  much ;  but  the  extra  force  comes 
from  the  reaction  of  the  walls  downward. 
The  liquid  exerts  a  thrust  perpendicular  to 
the  walls,  consequently  there  is  an  equal 
reaction  against  the  liquid ;  and  in  this  case 
there  is  a  component  vertically  downward. 

90,  If,  then,  the  same  liquid  is  standing  in  a  series  of 
connecting  tubes  of  any  shape,  the  level  must  be  the  same 
in  them  all.  Let  the  base  of  the  tubes  be  horizontal.  The 
pressure  must  be  the 
same  at  all  points  in 
the  liquid  along  this 
base ;  for,  if  it  was 
not,  there  would  be  - 
a  flow.  But  the  pres- 
sure below  each  tube, 
due  to  the  liquid  it- 


self, is  pgh-  and  as  FlG  gl 

p  and  g  are  the  same 

for  all  the  tubes,  h  must  be  also.  That  is,  the  free  surfaces 
must  be  at  the  same  height  above  the  horizontal  base ; 
and  so  they  are  all  in  the  same  level.  Also,  taking  a  series 
of  points  in  each  tube  at  the  same  depth  below  the  sur- 
faces, they  will  all  be  in  the  same  level ;  and  the  pressures 
are  the  same  for  all  the  points.  Hence,  in  connecting 
tubes  containing  the  same  liquid,  the  pressures  at  all 
points  in  the  same  level  are  the  same. 

Another  effect  of  the  influence  of  weight  upon  liquids  is 
that,  if  a  number  of  liquids  which  do  not  mix  or  react  on 
each  other,  and  which  are  of  different  densities,  such  as 


91]       PROPERTIES  OF  SIZE  AND  SHAPE  OF  MATTER      115 

mercury,  oil  and  water,  are  placed  in  a  vessel,  they  will 
arrange  themselves  so  that  the  surfaces  between  them  are 
horizontal,  and  in  such  an  order  that  the  densities  decrease 
from  layer  to  layer  as  one  goes  from  the  bottom  up. 

If  a  liquid  is  contained  in  a  vessel  of  any  shape,  there  is 
a  thrust  against  the  walls  due  to  the  pressures  at  each 
point.  The  pressure  at  the  free  surface,  due  to  the  liquid 
itself,  is  zero ;  that  at  any  other  point  at  a  depth  h1  is 
pghr\  so  that  the  pressure  increases  uniformly,  and  the 
average  pressure  in  a  liquid  of  depth  h  is  ^  p  g  h.  The  thrust 
or  force  against  the  wall  is  the  resultant  of  all  the  indi- 
vidual forces  acting  on  each  point  of  the  wall;  and  the 
point  of  application  of  this  resultant  is  called  the  "  centre 
of  pressure."  Its  position  can  generally  be  determined  by 
calculation,  if  the  shape  of  the  wall  is  known.  If  the 
wall  is  straight  and  has  the  same  width  from  top  to  bot- 
tom, this  resultant  thrust  is  the  average  pressure  multiplied 
by  the  area  of  the  wall. 

91.  Measurement  of  Density  of  a  Liquid.  If  different 
liquids  which  do  not  mix  or  react  on  each  other  are  placed 
in  connecting  tubes,  their  free  sur- 
faces will  not  stand  at  the  same 
height.  Let  two  such  liquids  of 
density,  p  and  plt  be  placed  in  the 
two  arms  of  a  TJ-tube.  The  denser 
liquid  will  occupy  the  lower  con- 
necting tube,  and,  if  there  is  enough 
of  it,  will  rise  to  a  certain  height 
in  one  arm,  while  the  lighter  liquid 
'will  be  in  the  other  arm.  Imagine  FlG  82 

a  horizontal  plane  passed  through 

the  two  arms  so  as  to  coincide  with  the  plane  where  the 
two  liquids  meet.  All  points  of  the  liquid  in  both  arms, 
which  are  in  this  plane,  must  have  the  same  pressure,  be- 
cause they  are  all  points  in  the  same  liquid,  viz.  the  denser 
one.  But  if  the  vertical  height  of  the  free  surface  of  one 


116 


THEORY   OF  PHYSICS 


[CH.  IV 


liquid  above  this  plane  is  hlt  the  pressure  in  this  plane  due 
to  it  is  />'  g  Aj  if  //  is  the  density  of  that  liquid ;  and,  if  the 
vertical  height  of  the  free  surface  of  the  other  liquid  above 
this  same  plane  is  h,  the  pressure  due  to  it  is  p  g  h. 


Hence 
or 


(10) 


Consequently,  if  the  density  of  one  liquid  is  known,  that 
of  the  other  may  be  determined  by  measuring  h  and  h^ 
The  density  of  water  at  4  °  C  is  1  (see  Art.  8)  ;  so  that  if 
water  is  used  as  one  of  the  liquids,  the  density  of  any 
liquid  which  does  not  mix  or  react  with  it  may  be  deter- 
mined by  this  method. 

If  the  two  liquids  do  mix  or  react,  a  simple  modifica- 
tion of  the  experiment  will  obviate  the  difficulty.     Invert 

the  TJ-tube  and  place  the  open 
ends  in  two  vessels  which  contain 
the  two  liquids  of  density  p  and 
Pi.  By  an  opening  in  the  con- 
necting tube  at  the  top,  exhaust 
some  of  the  air  in  the  tubes  until 
its  pressure  is  p.  The  liquids 
will  rise  in  the  tubes  to  vertical 
heights  h  and  hi  above  their  free 
surfaces.  At  the  free  surface  of 
each  liquid,  in  the  open  vessels, 
and  therefore  at  these  levels  in- 
side the  tubes,  the  pressure  is  that 
of  the  atmosphere.  But  in  the 
tube  containing  the  liquid  of  den- 
sity p,  this  pressure  is  due  to  a  pressure,  p,  on  the  top  of 
the  column  of  liquid  plus  the  pressure  pgh  due  to  the 
liquid  itself.  Hence 


m 


FIG.  83. 


Pressure  of  atmosphere  =  p  +  p  g  h . 


92]        PROPERTIES  OF  SIZE  AND  SHAPE  OF  MATTER       117 

Similarly  in  the  other  tube 

Pressure  of  atmosphere  (=  p  +  pig  hi. 
Hence  p  g  li  —  ft  g  h^ 

or  p  :  PI  =  J\  :  h .' 

So  that,  if  water  at  4°  C.  is  used  as  one  of  the  liquids,  or  if 
any  liquid  having  a  known  density  is  used,  the  density  of 
any  other  liquid  may  be  at  once  determined. 

In  both  of  these  experiments  the  tubes  which  contain 
the  liquids  must  be  so  large  that  there  is  no  capillary  effect 
(see  Art.  96) ;  but  there  is  no  other  restriction  as  to  their 
size  or  shape. 

92.  Archimedes'  Principle.  If  a  liquid  is  not  flowing,  any 
sensible  portion  of  it  is  at  rest,  and  may  be  considered  as 
enclosed  in  a  thin  film  so  as  to  be  separated 
from  the  rest  of  the  liquid.  The  weight  of 
this  portion  tends  to  make  it  move  down- 
ward, but,  since  it  does  not  move,  there 
must  be  an  equal  and  opposite  force  acting 
upward.  This  upward  force  is  the  result- 
ant of  all  the  pressures  acting  at  each 


point  of  the  film,  perpendicular  to  the  sur-          FiG  84 
face  of  the  film.     Consequently,  the  result- 
ant of  all  these  surface  pressures  on  the  film  is  a  force 
vertically  upward,  equal  to  the  weight  of  the  liquid  inside 
and  passing  through  its  centre  of  gravity.     It  is  supposed, 
of  course,  that  the  bottom  of  this  film  is  not  on  the  bottom 
of  the  vessel ;  for,  in  that  case,  there  would  be  no  pressure 
up  due  to  the  liquid. 

Let  this  portion  of  the  liquid  be  so  chosen  that  it  has 
the  same  shape  and  size  as  some  solid  which  is  denser 
than  the  liquid,  and  which  is  not  acted  upon  by  it.  Now, 
if  the  liquid  inside  the  film  be  removed,  and  its  place 
taken  by  the  solid,  the  surface  pressures  over  the  film 
have  not  been  changed,  because  the  film  has  not  changed 
in  the  least.  So,  while  the  force  down  on  this  solid  is 


118  THEORY   OF  PHYSICS  [CH.  IV 

its  weight,  there  is  also  a  force  up  equal  to  the  weight 
of  the  liquid  which  has  been  displaced,  if  the  bottom  of 
the  solid  does  not  rest  on  the  bottom  of  the  vessel.  This 
fact  is  known  as  "  Archimedes'  Principle."  So,  if  the  solid 
is  hung  from  one  pan  of  a  chemical  balance,  it  will  be 
noticed  that,  when  it  is  immersed  in  the  liquid,  there  is 
less  force  pulling  it  down.  This  "  loss  in  weight "  is  due 
to  the  force  upward  which  equals  the  weight  of  the  liquid 
displaced. 

93.  Measurement  of  Density  of  a  Solid.  Let  the  volume 
of  the  solid  be  v,  its  density  p ;  then  its  mass  is  p  v,  and  its 
weight  pvg.  When  it  is  totally  immersed  in  the  liquid, 
the  volume  of  the  liquid  which  is  displaced  is  also  v ;  and, 
if  its  density  is  p',  its  weight  is  p  v  g.  This  is  the  force 
upward,  if  the  solid  does  not  rest  on  the  bottom  of  the 
vessel ;  and  it  may  be  measured  by  weighing  the  solid  first 
in  air  (strictly,  in  a  vacuum),  then  when  immersed  in  the 
liquid.  The  weight  in  air  (strictly,  in  a  vacuum)  is  p  v  g  ; 
the  difference  in  the  two  weighings,  or  the  "  loss  in  weight," 
is  pr  v  g.  The  ratio  of  these  two  quantities  is  p  /  p' ;  and  so, 
if  the  density  of  the  liquid  is  known,  that  of  any  solid 
denser  than  it,  and  which  is  not  acted  upon  by  it,  may  be 
determined.  Water  at  4°  C.  is  generally  the  liquid  used, 
because  its  density  is  1.  If  a  solid  has  a  density  less  than 
that  of  the  liquid,  it  may  be  immersed  in  the  liquid  by 
hanging  some  very  heavy  solid  below  it,  and  then  keep- 
ing this  last  solid  immersed  in  the  liquid  during  both 
weighings. 

This  principle  also  gives  a  method  by  which  the  den- 
sities of  two  liquids  may  be  compared.  If  the  solid  is 
immersed  in  turn  in  the  two,  its  loss  in  weight  in  them 
will  be  pigv  and  pzgv  respectively;  which  gives  at  once 
the  ratio  of  the  two  densities. 

Since,  then,  a  liquid  produces  an  upward  force  upon  any 
immersed  solid,  which  does  not  rest  on  the  bottom,  there 
must  be  an  equal  reaction  of  the  solid  upon  the  liquid. 


94]       PROPERTIES  OF  SIZE  AND  SHAPE  OF  MATTER       119 

So  that,  if  the  vessel  containing  the  liquid  stands  on  a  plat- 
form-balance, there  will  be  an  additional  force  down,  which 
will  be  registered  on  the  balance,  when  a  solid  is  held 
immersed  in  it,  but  not  touching  the  bottom.  This  addi- 
tional force  equals  the  weight  of  the  liquid  displaced.  So 
a  platform-balance  may  be  used,  as  well  as  a  chemical  bal- 
ance, for  the  measurement  of  densities. 

Other  instruments,  such  as  Hydrometers,  Jolly's  Spring 
Balance,  etc.,  have  been  devised  and  are  in  daily  use  for 
these  same  measurements. 

94.  Floating  Bodies.  If  a  body  floats  on  the  surface  of  a 
liquid,  it  is  in  equilibrium ;  and 
so  the  forces  in  any  direction  must 
be  balanced.  The  force  down  is 
its  weight,  and  it  acts  in  a  line 
passing  through  the  centre  of 
gravity  of  the  body.  The  force 
up  equals  the  weight  of  the  liquid 
displaced,  and  passes  through  the 
centre  of  gravity  of  this  liquid. 

So,  when  a  body  is  floating,  it  displaces  its  own  weight  of 

the  liquid  ;  and  the  centre  of 
gravity  of  the  body  must  lie  in 
the  same  vertical  line  as  that  of 
the  liquid  displaced. 

This  equilibrium  may  be  stable, 
unstable,  or  neutral.  The  question 
must  be  investigated  by  consider- 
ing what  happens  when  the  body 
is  slightly  tipped  one  side.  When 
the  body  is  in  equilibrium,  the 
two  centres  of  gravity,  that  of  the 
body  and  that  of  the  displaced 
liquid,  must  lie  in  the  same  verti- 
cal line,  which  is  called  the  "  axis."  Draw  this  line  in  the 
body,  and  consider  it  fixed  there.  Now,  if  the  body  is 


FIG.  85. 


FIG.  86. 


120  THEORY   OF  PHYSICS  [CH.  IV 

tipped,  the  axis  will  be  inclined  to  the  vertical ;  it  will 
still  pass  through  G,  the  centre  of  gravity  of  the  body,  be- 
cause both  it  and  the  axis  are  fixed  in  the  body ;  but  the 
centre  of  gravity  of  the  displaced  liquid,  (7,  will  now  lie  to 
one  side.  So  there  is  a  couple  acting  on  the  body,  made 
up  of  the  two  equal  forces,  one  down  through  Gr,  the  other 
up  through  0.  Let  a  line  drawn  vertically  through  C  meet 
the  axis  in  a  point  m,  called  the  "  metacentre."  If  m  lies 
above  G  on  the  axis,  it  is  evident  that  the  couple  acting  on 
the  body  tends  to  restore  the  axis  to  its  vertical  position  ; 
and  so  the  equilibrium  was  stable.  While,  if  m  is  below 
G  on  the  axis,  the  couple  will  tend  to  make  the  body  tip 
farther ;  and  the  equilibrium  was  unstable.  The  position 
of  the  metacentre  depends  upon  the  shape  and  construction 
of  the  body  and  upon  the  way  it  is  displaced ;  and,  in  mak- 
ing calculations  to  determine  it,  the  tipping  is  assumed  to 
be  very  slight.  Any  ordinary  boat  has  two  principal  rneta- 
centres,  —  one  for  a  rolling  motion,  the  other  for  a  pitching 
one. 

95.  Work  Required  to  Change  the  Volume  of  a  Liquid.  As 
a  consequence  of  the  equality  of  pressure  throughout  a 
liquid,  it  is  not  difficult  to  calculate  how 
much  work  is  required  to  increase  the  vol- 
ume of  a  liquid.  (This  increase  may  be 
produced  by  forcing  in  more  liquid  or  by 
raising  the  temperature.)  Imagine  the  liq- 
uid enclosed  in  a  cylinder,  into  one  end  of 
which  is  fitted  a  plane  movable  piston.  Let 
the  area  of  this  piston  be  A,  and  the  pres- 


FIG.  87.  sure  of  the  liquid  on  it  p.  Then  the  force 
is  p  A  ;  and,  if  the  piston  is  driven  out  a 
distance  I  owing  to  the  increase  in  volume,  the  work  done 
is  pAl,  if  the  pressure  remains  constant.  But  A  I  is  the 
increase  in  volume ;  and  so  the  work  done  at  constant 
pressure  is  the  product  of  the  pressure  and  the  change  in 
volume.  (If  during  the  change  in  volume  the  pressure 


96]        PROPERTIES  OF  SIZE  AND  SHAPE  OF  MATTER       121 


changes  uniformly,  the  average  pressure  is  the  mean  of  the 
initial  and  final  values  ;  and  the  work  done  is  the  product 
of  the  average  pressure  and  the  change  in  volume.) 


work  =  p  (v2  — 


(11) 


96.  Liquids  in  Motion.  If  a  liquid  which  had  no  viscosity 
was  allowed  to  escape  through  an  opening  in  a  thin  wall 
at  the  depth  h  below  its  free  surface,  its  speed  of  efflux 
would  be 

*  =  V2jh (12) 

Because,  owing  to  the  fact  that  there  is  in  this  case  no 
resistance  to  the  motion  of  one  portion  of  the  liquid  over 
another,  a  particle  of  the  liquid 
as  it  escapes  will  have  the  same 
speed  as  if  it  had  fallen  freely 
from  the  free  surface  through  the 
height  h.  (See  Art.  18.)  The 
pressure  at  a  point  just  outside 
the  opening  is  that  of  the  atmos- 
phere, while  that  just  inside  is 
greater  than  this  by  an  amount 
p  g  h.  So  if  p  is  the  difference  of 
pressure  outside  and  in  at  the  opening, 


FlG  88 


p  =  p  g  h ;  and  so  s  — 


(13) 


All  actual  liquids  are  viscous ;  and  so  the  observed  speeds 
of  efflux  are  always  less  than  this.  If  the  liquid  escapes 
through  a  long  tube,  it  is  found  by  experiment  that  the  ma- 
terial of  the  tube  has  no  effect  on  the  rate  of  efflux.  This 
proves  that  there  is  a  layer  of  the  liquid  formed  over  the 
inner  wall  of  the  tube,  and  that  this  does  not  move,  the 
friction  being  then  between  the  layers  of  the  liquid,  and 
depending  only  upon  the  viscosity  of  the  liquid. 


122 


THEORY  OF  PHYSICS 


[CH.  IV 


If  two  liquids  are  placed  directly  in  contact,  there  is  a 
gradual  diffusion  of  one  into  the  other,  which  goes  on 
entirely  independently  of  their  relative  densities.  In  time 
the  mixture  would  be  uniform. 

Certain  membranes  and  porous  materials  such  as  parch- 
ment, unglazed  earthenware,  etc.  have  the  property  of 
being  permeable  for  some  substances  and  not  for  others. 
This  general  phenomenon  of  the  passage  of  substances 
though  semi-permeable  membranes  has  received  the  name 
"osmosis."  If  a  miscellaneous  mixture  of  a  great  many 
substances  is  placed  in  a  parchment  dish  and  floated  on 
water,  it  is  noticed  that  after  a  certain  time  some  of  the 
substances  have  passed  through  into  the  water,  while  others 
never  will. 

If  a  solution  of  cane-sugar  in  water  is  enclosed  in  a 
tube  over  one  end  of  which  is  a  parchment  cap,  and  if 
this  end  is  dipped  into  pure  water? 
it  is  observed  that  the  water  passes 
through  the  parchment  into  the  tube. 
So  that  the  level  of  the  liquid  inside 
the  tube  becomes  higher  than  it  is 
outside  by  an  amount  A,  when  the 
flow  stops ;  that  is,  it  takes  a  pressure 
pgh  to  overcome  the  tendency  of  the 
water  to  come  in.  This  tendency  is 
caused  by  the  sugar  in  solution ;  and 
the  pressure  p  g  h  is  called  the  "  os- 
motic" pressure  of  the  solution. 

In  the  flow  of  a.  liquid  through  a 

tube  of  any  shape,  if  the  speed  is  greater  at  one  point 
than  at  another,  the  pressure  at  the  first  point  must  be 
less  than  at  the  other ;  because  to  produce  an  increase 
of  speed  a  force  is  necessary,  that  is,  a  fall  in  pressure. 
This  is  the  explanation  of  a  ball  in  a  fountain  .remain- 
ing poised,  of  the  so-called  "  ball-nozzle,"  and  several  other 
phenomena. 


FIG.  89. 


97]        PROPERTIES  OF  SIZE  AND  SHAPE  OF  MATTER       123 


FIG.  90. 


97.  Capillarity.  The  property  of  a  liquid  which  dis- 
tinguishes it  from  a  gas  is  that  it  has  a  definite  size, 
and  so  is  separated  from  other  liquids  or  gases  by  a  defi- 
nite surface  unless  there  is  diffusion.  As  a  consequence 
of  a  liquid  having  this  surface,  it  has  certain  character- 
istics, and  obeys  certain  laws. 

It  is  at  once  seen  that  a  particle  of  a  liquid  in  its 
free  surface  is  in  a  different  condition  from  a  particle  in 
the  interior ;  for  the  latter  is  influenced 
by  particles  of  the  liquid  on  all  sides  of 
it,  while  the  one  in  the  surface  is  influ- 
enced by  only  half  as  many.  The  funda- 
mental property  of  a  liquid  surface  is  its 
tendency  to  contract  and  so  become  as 
small  as  possible.  This  is  proved  by  a 
great  many  experiments.  A  mixture  of  water  and  alco- 
hol can  be  made  which  has  the  same  density  as  olive  oil. 
So,  if  a  drop  of  oil  is  put  in  a  vessel  of  this  mixture,  it 
will  stand  anywhere  in  the  water  entirely  uninfluenced  by 
the  force  due  to  the  earth.  Now,  it  will  be  found  that 
this  drop  will  have  a  spherical  form ;  and  a  sphere  has  the 
least  surface  for  a  given  volume.  Ordinary  drops  of  water 

and  other  liquids  are  not 
spherical,  because  their 
shape  is  changed  owing  to 
the  earth's  force  of  gravi- 
tation. 

Again,  if  a  film  of  a  soap- 
mixture  -stretched  across 
a  circular  wire  frame  is 
broken  at  any  point,  it 
will  instantly  contract. 
The  following  experiment 
illustrates  the  same  fact.  Fasten  a  string  to  one  point  of 
the  wire  frame,  and,  holding  the  other  end  of  the  string, 
dip  the  frame  in  a  soap-mixture  so  as  to  make  over  it  a  film. 


FIG.  91. 


124  THEORY   OF  PHYSICS  [CH.  IV 

in  which  the  string  lies.  Then  break  the  film  on  one 
side  of  the  string,  and  the  film  will  contract  until  the 
stretched  string  forms  the  arc  of  a  circle ;  for  then  the  film 
is  as  small  as  it  can  be  under  the  conditions.  If  the  string 
is  loosened  slightly,  the  film  contracts  still  more ;  and,  if 
it  is  desired  to  stretch  the  film,  force  must  be  exerted  on 
the  string. 

So  it  is  proved  that  there  are  forces  acting  in  any  liquid 
film,  tending  to  make  it  contract.  If  a  line  is  imagined 
drawn  on  the  surface  of  a  liquid,  there  is  a  certain  force 
acting  across  it  holding  together  the  portions  of  the 
surface  on  its  two  sides.  The  force  acting  across  one 
centimetre  of  any  line  in  a  surface  is  called  the  "surface- 
tension,"  T. 

Thus,  consider  any  line  1  cm.  long  in  the  film,  as  shown  ; 
there  are  forces,  T,  in  both  directions  perpendicular  to  the 
line.  Since  they  are  equal,  the  line 
does  not  move ;  but,  if  the  surface- 
tension  on  one  side  is  weakened  in 
any  way,  then  the  line  will  move 
towards  the  other  side.  This  may 


F      92  be  shown  by  heating  the  film  on 

one  side  or  by  putting  some  im- 
purity in  it ;  for  the  surface-tension  is  diminished  by  both 
these  causes.  This  explains  the  rapid  motion  given  to  a 
bit  of  camphor  or  sodium  when  tossed  upon  a  surface  of 
water.  The  solid  dissolves  unequally  at  various  points; 
so  the  surface  is  rendered  impure  at  some  points  faster 
than  at  others ;  and  the  surface-tension  being  thus  weak- 
ened more  at  these  points,  there  is  motion  in  the  opposite 
direction. 

The  surface-tension  of  a  pure  liquid  in  contact  with  any 
definite  substance  is  constant  for  a  given  temperature  ;  for, 
when  the  surface  is  increased,  there  is  not  a  stretching  of 
the  surface,  but  the  formation  of  more  surface  of  the  same 
liquid.  For  this  reason  a  vertical  film  of  a  pure  liquid 


97]       PROPERTIES  OF  SIZE  AND  SHAPE  OF  MATTER       125 

cannot  be  made  here  on  the  surface  of  the  earth  ;  for,  owing 
to  the  weight  of  the  liquid  in  a  vertical  film,  the  forces 
must  be  greater  at  the  top  than  at  the  bottom. 

Owing  to  this  tendency  of  a  surface  to  contract,  a  curved 
surface  tends  always  to  have  a  smaller  radius  ;  and  so  there 
is  a  pressure  produced  toward  the  centre  of  curvature. 
Thus,  a  bubble  will  contract  until  stopped  by  the  pressure 
of  the  gas  inside.  A  drop  contracts  until  stopped  by  the 
pressure  of  the  liquid.  It  is  not  difficult  to  find  a  relation 
between  this  pressure  of  the  liquid  in  a  drop  and  the  sur- 
face-tension. Imagine  the  drop  cut  into  two  hemispheres, 
and  then  replaced.  They  will  be  held  together  by  the 
forces  acting  across  the  equator  where  the  cut  is  made  in 
the  surface.  If  r  is  the  radius  of  the  sphere,  the  length  of 
the  equator  is  2  TT  r  ;  and,  as  T  is  the  force  across  one  centi- 
metre, the  entire  force  holding  the  two  hemispheres  to- 
gether is  2  TT  r  T.  As  a  consequence  of  this  force,  the 
hemispheres  are  pressed  together  until  a  pressure  p  is 
reached  which  counterbalances  the  force.  The  area  of  the 
equatorial  section  is  TT  r2  ;  and,  as  p  is  the  force  per  unit 
area,  the  entire  reaction  of  the  two  hemispheres  against 
each  other  is  TT  r2  p.  Hence,  since  action  and  reaction  are 
equal  but  opposite, 


and  so  p  =  2  T  /  r  ......     (14) 

Therefore  a  surface  whose  tension  is  T  and  whose  radius 
of  curvature  is  r  produces  a  pressure  towards  the  centre  of 
curvature  equal  to  2  T/r. 

If  the  surface,  instead  of  being  a  sphere,  had  been  a 
cylinder  of  radius,  r,  the  pressure  would  have  been 


(IB) 


If  certain  solids  are  immersed  in  certain  liquids  and 
then  taken  out,  they  are  found  to  be  covered  with  a  film 


126 


THEORY   OF  PHYSICS 


[CH.  IV 


FIG.  93. 


of  the  liquid,  and  are  said  to  be  "  wet "  by  it.  Thus,  glass 
is  wet  by  water.  Other  substances  are  not  wet ;  e.  g.  glass 
is  not  wet  by  mercury  under  ordinary  conditions. 

If  a  plate  of  glass,  previously  mois- 
tened, is  dipped  in  water,  the  film  of 
the  water  is  at  first  rectangular,  being 
part  on  the  glass  and  part  on  the 
surface  of  the  water.  But  this  film 
tends  to  contract;  and  so  the  corner 
is  rounded  off,  and  the  water  is  said 
to  rise  against  the  glass.  The  con- 
traction of  the  film  will  continue  until 
stopped  by  the  weight  of  the  liquid  raised.  Let  a  glass 
tube  of  small  bore,  previously  moistened  inside,  be  dipped 
into  a  vessel  of  water.  The  water  will  rise  against  the 
glass,  and  in  the  interior  the  water 
will  rise  to  a  considerable  height  if 
the  bore  is  small  enough.  For  the 
liquid  film  inside  the  tube  is  over  the 
walls  of  the  tube  and  over  the  water 
where  the  tube  enters  the  surface ; 
that  is,  it  is  like  the  finger  of  a  glove. 
This  surface  tends  to  contract ;  and  it 
can  do  so  if  the  liquid  surface  inside 
the  tube  rises  ;  so  it  will  continue  to 
rise  until  stopped  by  the  weight  of  the 
liquid  raised.  Since  the  water  .wets 
the  glass,  the  surface  of  the  liquid  in- 
side the  tube  is  spherical,  if  the  tube  is 
circular,  and  the  radius  of  the  sphere 
equals  the  radius  of  the  tube.  The 
pressure  at  a  point  of  the  level  sur- 
face of  the  water  outside  the  tube  is 
that  due  to  the  atmosphere ;  so  it  must  be  this  also  at  the 
same  level  inside  the  tube.  But  the  pressure  there  is  due 
to  three  causes,  —  the  pressure  of  the  atmosphere  on  top  of 


FIG.  94. 


97]        PROPERTIES  OF  SIZE  AND  SHAPE  OF  MATTER       127 

the  liquid  column,  the  pressure  upward  due  to  the  curved 
surface,  the  pressure  due  to  the  height  of  the  column. 
Call  this  vertical  height  h,  and  the  radius  of  the  tube  at' 
the  point  where  the  curved  surface  is,  r.  Then, 

Atmospheric  pressure  = 

atmospheric  pressure  —  2  T  /r  +  pgh. 
Hence  pgh  =  2T/r; 


2  T 
and  so  h  =  -       ......     (16) 

pgr 

Consequently  the  vertical  height  of  the  column  varies  in- 
versely as  the  radius  of  the  tube  at  the  point  where  the 
curved  surface  is,  but  is  independent  of  the  shape  or  size 
of  the  tube  elsewhere,  because  the  pressure  in  a  liquid  de- 
pends only  on  the  depth  of  the  liquid,  not  on  the  shape  of 
the  containing  vessel.  So,  if  h,  p,  g,  r  are  measured,  T  may 
be  calculated.  T  will  vary  with  the  temperature,  always 
decreasing  as  the  temperature  rises  ;  it  will  also  vary  with 
the  substance  with  which  the  liquid  is  in  contact.  These 
phenomena  due  to  the  surface-tension  were  first  observed 
in  the  rise  of  liquids  in  small  tubes,  and  so  are  often  called 
"  capillary  "  phenomena,  because  the  bore  of  the  tubes  is 
comparable  to  the  size  of  a  hair,  the  Latin  word  for  hair 
being  "  capillus." 

In  identically  the  same  manner  as  this,  it  may  be  ex- 
plained why,  if  a  glass  tube  is  dipped  into  a  vessel  of 
mercury,  the  mercury  inside  the  tube  sinks. 

Since  in  any  curved  surface  p  —  2  T  /  r,  it  would  require 
an  infinite  pressure  to  make  a  surface  of  radius  0  ;  and  the 
-smaller  the  radius,  the  greater  the  pressure.  This  is  why 
the  formation  of  drops  or  bubbles  is  greatly  facilitated  by 
the  presence  of  points  or  nuclei;  for,  as  they  begin  to 
form,  the  curvature  of  the  surface  is  that  of  the  point  or 
nucleus,  and  so  is  of  finite  dimensions.  This  has  a  most 
important  bearing  on  the  condensation  of  any  vapor,  and 
also  on  the  boiling  of  liquids. 


128 


THEORY   OF  PHYSICS 


[CH.  IV 


FIG.  95. 


A  soap-bubble  has  two  films  of  nearly  the  same  radius ; 

so  the  pressure  of  the  gas  inside  must  be  4  T I  r. 

When  two  solids  which  are  wet  by  a  liquid  are  dipping 

in  the  liquid  and  are  brought  so  near  that  the  liquid  rises 
between  them,  there  will  be  a  motion  of 
the  two  solids  towards  each  other,  be- 
cause in  between  the  two  plates  at 
points  above  the  level  of  the  liquid  out- 
side the  pressure  is  less  than  the  atmos- 
pheric pressure  ;  and  so  the  solids  are 
pressed  together  by  the  air  outside. 
Similarly,  it  may  be  shown  that,  if  the 
two  solids  are  not  wet  by  the  liquid, 
they  will  also  be  pressed  together  when 
they  are  dipping  in  the  liquid.  But  two 
solids,  one  of  which  is  wet  by  the  liq- 
uid, the  other  not,  are  pushed  apart,  if 
they  are  dipping  in  the  liquid  near  each 
other. 
If  a  drop  of  a  liquid  is  placed  in  a  conical  tube,  it  may 

move  in  one  direction  or  the  other,  depending  upon  the 

different  curvatures  of  the  surfaces  at  its  two  ends. 

SPECIAL  PROPERTIES  OF  GASES 

98.  Introduction.     The  distinguishing  properties  of  a  gas 
are :  1.  Any  force,  no  matter  how  small,  can  cause  one  por- 
tion to  move  over  another ;  2.  It  has  no  definite  size  of  its 
own,  but  fills  the  space  enclosing  it.     As  a  consequence  of 
the  first  property,  all  the  laws  of  pressure  which  were  de- 
rived for  a  liquid  hold  equally  well  for  a  gas ;  but,  as  the 
gas  has  no  free  surface,  and  can  have  its  volume  changed 
most  easily,  it  has  certain  new  properties  in  connection 
with  its  volume ;   3.  The  molecules  of  a  gas  are  in  rapid 
motion  in  all  directions,  on  the  kinetic  theory. 

99.  Properties  of  a  Gas  at  Rest.     Without  further  proof, 
then,  the  following  statements  are  true:  — 


129 


UNIVERSITY  OF 

1001     PROPERTIES  OF  SIZE  AND  SMPE*  OF- 
TFEPARTP/iENT  OF  PHYSICS 

1.  The  pressure  due  to  a  gas  is  always  perpendicular  to 
any  surface  in  contact  with  it. 

2.  The  pressure  at  any  point  in  a  gas  is  the  same  in  all 
directions. 

3.  The  pressure  at  any  point   in  a  gas  is  due  to  two 
causes  :  (a)  the  pressure  of   the  enclosing  vessel ;  (b)  the 
pressure  due  to  the  weight  of  the  gas  above  the  point. 

4.  Archimedes'    Principle    applies   perfectly    to    gases. 
That  is,  a  solid  immersed  in  a  gas  is  buoyed  up  by  a  force 
which  equals  the  weight  of  the  gas  displaced.    So,  in  weigh- 
ing solids  in  the  air,  a  correction  must  be 

made  for  the  buoyancy  of  the  atmosphere. 

100.  Pressure  due  to  Weight  of  Atmosphere. 
Unless  a  column  of  gas  is  very  high,  it  pro- 
f  duces  a  comparatively  small  pressure ;  for,  if 
the  density  of  the  gas  is  p,  the  difference  in 
pressure  between  two  points  at  a  vertical 
height  h  apart  is  p  g  h.  p  is  very  small  for 
gases  ;  so  h  must  be  large,  if  this  pressure  is 
to  be  sensible.  An  illustration  of  a  high  col- 
umn of  a  gas  is  afforded  by  the  earth's  at- 
mosphere, which  does  produce  a  large  pressure. 
It  may  be  measured  by  balancing  it  against  a 
column  of  a  liquid.  Completely  fill  with 
mercury  a  glass  tube,  which  is  closed  at  one 
end,  of  rather  large  cross-section,  and  about 
80  cm.  long  ;  invert  it,  allowing  no  air  to 
enter ;  and  place  the  open  end  in  a  basin  of 
mercury.  It  will  be  observed  that  the  mer- 
cury will  continue  to  stand  in  the  tube  up  to 
a  certain  height  above  the  free  surface  in  the 
basin.  The  pressure  must  be  the  same  at 
the  free  surface  of  the  mercury  as  at  the  same  level  in  the 
tube.  The  pressure  outside  is  the  atmospheric  pressure 
due  to  the  air;  that  inside  is  pgh,  where  p  is 'the  density 
of  the  mercury  and  h  is  the  height  of  the  column.  This 


FIG.  96. 


130  THEORY  OF  PHYSICS  [CH.  IV 

is  the  entire  pressure,  because  there  is  no  gas  pressing 
down  on  the  top  of  the  column,  there  being  a  vacuum 
above  the  column,  except  for  the  slight  amount  of  mercury 
vapor  which  may  arise  from  the  liquid  mercury.  Hence 

atmospheric  pressure  =  p  g  h     .     .     .     (17) 

h  varies  from  hour  to  hour,  but  is  always  in  the  neighbor- 
hood of  76  cm.  for  mercury.  Such  an  instrument  as  this, 
which  measures  the  pressure  of  the  atmosphere,  is  called  a 
barometer.  The  height  to  which  the  liquid  stands  does 
not  depend  on  the  shape  or  size  of  the  tube ;  for  it  is  the 
pressure  which  determines  the  height.  At  different  tem- 
peratures, however,  the  density  of  the  liquid  is  different ; 
and  so,  even  though  the  pressure  is  unchanged,  the  height  h 
may  vary.  Consequently,  to  compare  pressures  by  the 
heights  of  the  barometric  column,  these  must  be  given  in 
terms  of  some  standard  liquid  which  does  not  change. 
This  is  done  by  "correcting"  the  actual  reading  on  the 
barometer  in  a  manner  explained  in  Heat  (Art.  175),  by 
calculating,  from  the  observed  temperature  and  height, 
that  height  to  which  the  column  would  have  risen  if  the 
density  of  the  liquid  were  that  which  it  has  at  0°  C. 

There  are  various  forms  of  barometers,  and  various 
liquids  are  used ;  but  in  each  the  pressure  is  given  by 
pgh,  where  p  is  the  density  of  the  liquid,  and  li  is  the 
height  from  the  free  surface  to  the  top  of  the  column. 
Thus,  if  the  liquid  is  mercury,  whose  density  is  13.6,  and 
if  the  height  is  76  cm.,  the  atmospheric  pressure  is  13.6  X 
980  X  76  =  1013000  dynes  per  sq.  cm. 

There  are  innumerable  illustrations  of  this  pressure  of 
the  air.  If  two  metal  hemispheres  are  carefully  fitted  to- 
gether, and  the  air  is  pumped  out  from  the  interior,  a  great 
force  is  necessary  to  pull  them  apart,  because  the  pressure 
of  the  air  holds  them  together.  If  a  narrow  goblet  is 
filled  with  water,  then  covered  with  a  card  and  carefully 
inverted,  the  water  will  not  escape. 


102]    PROPERTIES  OF  SIZE  AND  SHAPE  OF  MATTER       131 


101.  Pumps.     The  ordinary  lifting-pump  is  also  an  illus- 
tration of  the  pressure  of  the  air.    It  consists  of  a  cylinder  in 
which  moves  a  close-fitting  piston,  and  which  is  connected  at 
its  lower  end  by  a  pipe  with  the  well  or  sup- 
ply of  water ;  in  the  piston  at  B  and  at  the 

top  of  the  pipe  at  A  there  are  valves  opening 
upward.     When  the  piston  is  being  raised,  its 
valve,  B,  is  closed,  and  the  pressure  of  the  air       |g 
below  the  piston  is  so  diminished   that  the 
atmospheric  pressure  on  the  free  surface  in 
the  well  forces  water  up  through  the  pipe  and 
the  valve  A  into  the  cylinder.     Now,  if  the 
piston  is  forced  down,  the  valve  A  closes  and 
that  at  B  opens,  allowing  the  water  to  pass 
through  above  the  piston.     So,  when  the  pis- 
ton is  again  raised,  this  water  is  lifted  and 
runs  out  at  the  top  of  the  pump,  while  the 
cylinder  below  the  piston  becomes  again  full 
of  water.     And  so  the  process  continues.     Of 
course  the  pump  cannot  work  if  the  pipe  from       pIG  97 
the  well  is  so  long  that  the  atmospheric  pres- 
sure cannot  lift  the  water  that  high.     This  height  is  given 
by  the  condition  that  pgh  for  water  equals  the  atmos- 
pheric pressure.     If  this  is  76  cm.  of  mercury, 

1  x  g  X  h  =  13.6  X  g  X  76. 
Hence  the  limiting  height  h  is 

k  =  13.6  X  76  =  1033.6  cm. 

In  practice  this  height  can  never  be  reached  in  a  pump, 
owing  to  leaks  around  the  piston,  which  allow  some  air 
to  remain  below  it. 

102.  Siphon.     Another  illustration  of  atmospheric  pres- 
sure is  given  in  the  "  siphon,"  which  consists  of  a  tube 
bent  into  a  TT  with  one  arm  longer  than  the  other,  and 
which  is  used  to  make  a  liquid  flow  out  of  a  basin  over  its 


132 


THEORY   OF  PHYSICS 


[CH.  IV 


edge.  The  tube  is  filled  with  the  liquid,  and  is  then  placed 
with  its  shorter  arm  dipping  in  a  basin  of  the  liquid,  and 

its  longer  arm  projecting  down 
outside  the  basin.     To  do  this, 
the  lower  end,  B,  of  the  longer 
arm  must  be  kept  closed ;  and 
the  pressure  at  a  point,  A,  in 
-A     this  arm  at  the  level  of  the 
free  surface  in  the  basin  will 
be  at  the  atmospheric  pres- 
sure, because  of  the  equality 
of  pressure  at  the  same  level 
B       in    connecting    tubes    of    the 
FIG.  98.  same  liquid.     So,  if  now  the 

end  B  is  opened,  the  pressure 

of  the  liquid  between  B  and  A  will  be  in  excess  of  the 
atmospheric  pressure  outside,  and  the  liquid  will  start  to 
run  out ;  but,  as  it  does  so,  the  pressure  in  the  long  arm  is 
diminished,  and  the  atmospheric  pressure  on  the  surface  of 
the  liquid  in  the  basin  forces  over  some  liquid,  and  so  the 
stream  is  kept  up.  The  vertical  height  of  the  top  of  the 
siphon  above  the  free  surface  in  the  basin  must  not  be 
greater  than  corresponds  to  the  atmospheric  pressure,  or 
the  liquid  will  not  flow  out.  Hence  there  would  be'  no 
flow  at  all  (at  least  from  this  cause)  if  the  basin  and 
siphon  were  placed  in  a  vacuum. 

103.  Measurement  of  Pressure  of  a  Gas,  If  the  pressure 
of  the  atmosphere  is  known,  the  pressure  of  any  gas  en- 
closed in  a  vessel  may  be  measured.  Insert  in  the  side  of 
the  vessel  a  bent  tube  open  at  both  ends,  which  contains 
some  liquid,  e.  g.  mercury.  The  liquid  will  in  general 
stand  at  different  heights  in  the  two  tubes  ;  and,  if  the 
difference  in  the  two  levels  is  li,  the  pressure  of  the  gas 
inside  is  different  from  that  of  the  atmosphere  by  an 
amount  pg  h,  where  p  is  the  density  of  the  liquid.  If  the 
liquid  is  forced  up  in  the  open  tube  so  as  to  stand  higher 


103]     PROPERTIES  OF  SIZE  AND  SHAPE  OF  MATTER       133 


there  than  in  the  arm  connected  with  the  gas,  the  pressure 
of  the  gas  is  greater  than  that  of  the  atmosphere  by  the 
amount  p  g  h  ;  whereas,  if  the  contrary  is  true,  the  pressure 
of  the  gas  is  less  than  that  of  the  atmosphere  by  p  g  h. 
Such  an  apparatus  for  measuring  pressures  is  called  an 
open  manometer. 


FIG.  99. 


FIG.  100. 


If  a  gas  is  enclosed  in  the  space  above  a  column  of  mer- 
cury in  a  barometer  tube,  its  pressure  will  force  the  mercury 
down  a  certain  distance ;  and,  if  the  pressure  of  the  atmos- 
phere is  known,  that  of  the  gas  can  be  measured  at  once. 
Let  the  column  of  mercury  stand  at  the  height  h.  Then, 
equating  the  pressures  outside  and  inside  the  tube  at  the 
level  of  the  free  surface,  atmospheric  pressure  =  pgh  +  pres- 
sure of  gas  above  column,  and  so  pressure  of  gas  =  atmos- 
pheric pressure  —pgh. 


134  THEORY  OF  PHYSICS  [CH.  IV 

104.  Work  Necessary  to  Change  the  Volume  of  a  Gas.     As 
a  further  consequence  of  the  equality  of  pressure  through- 
out a  gas,  it  is  at  once  evident  that,  as  in  the  case  of  a 
liquid,  the  work  done  in  changing  the  volume  while  the 
pressure  is  constant  is  the  product  of  the  pressure  and  the 
amount  of  the  change  in  volume  (see  Art.  95). 

105.  Gases  in  Motion.     Just  as  a  liquid,  escaping  through 

a  thin  opening  under  a  pressure  p,  has  a  speed  s  =  y  — , 

P  - 

so  a  gas  under  the  same  conditions  obeys  the  same  law. 
The  pressure,  p,  which  produces  the  flow  of  the  gas  is  the 
difference  of  pressure  of  that  particular  gas  on  the  two 
sides  of  the  opening ;  and  the  density,  p,  is  that  of  the  gas. 
By  measuring,  then,  the  quantities  of  different  gases  which 
escape  in  a  given  time  under  the  same  conditions,  it  is  pos- 
sible to  determine  the  ratio  of  the  densities  of  any  two 
gases.  (Another  method  of  determining  densities  would 
be  to  weigh  a  large  bulb,  of  known  volume  v,  first  empty 
and  then  filled  with  the  gas.  If  the  difference  in  weight 
is  m  grams,  the  density  of  the  gas  under  the  existing  con- 

tn 
ditions    of  temperature   and   pressure   is    —  •)    Gases,  like 

liquids,  diffuse  into  each  other ;  so  that  in  time  the  mix- 
ture becomes  uniform.  Gases  also  pass  through  many 
solids,  and  are  often  absorbed  by  solids  and  liquids.  Since 
the  rate  of  escape  through  small  openings  varies  inversely 
as  the  square  root  of  the  density,  dense  gases  pass  through 
porous  bodies  much  more  slowly  than  do  rare  ones. 

If  a  gas  is  flowing  through  any  irregular  space,  the  pres- 
sure is  least  where  the  velocity  is  greatest,  just  as  is  the 
case  of  a  liquid.  This  explains  why,  if  a  jet  of  air  is 
blown  over  the  end  of  a  tube  the  other  end  of  which  dips 
into  a  liquid,  the  liquid  rises  in  the  tube  and  may  reach 
the  top  and  be  blown  away,  as  in  an  "  atomizer." 

106.  Properties  of  Gases  as  Distinct  from  Liquids.  The 
particular  property  of  a  gas  which  distinguishes  it  from  a 


108]     PROPERTIES  OF  SIZE  AND  SHAPE  OF  MATTER       135 

liquid  is  the  fact  that  it  takes  the  size  of  the  containing 
vessel.  Consequently  a  gas  is  very  compressible,  because 
the  volume  of  the  containing  vessel  may  be  changed  as 
one  wishes.  The  particles  of  a  gas,  i.  e.  its  smallest  por- 
tions, are  also  moving  about  with  a  great  freedom  of  mo- 
tion, even  more  than  is  the  case  with  a  liquid.  As  a 
consequence  of  these  facts,  there  are  certain  so-called 
laws  of  gases,  which  have  been  found  to  be  true  by  care- 
ful experiments. 

107.  Dalton's  Law.      This  states  that,  if   several   gases 
which  do  not  react  on  each  other  are  put  inside  a  cer- 
tain vessel,  the  mixture  becomes  uniform  throughout,  and 
the  pressure  at  any  point  is  the  sum  of  the  pressures  which 
each  gas  would  produce  if  it  occupied  the  vessel  by  itself. 
This  is  equivalent  to  saying  that  the  particles  of  each  gas 
are  moving  rapidly  in  all  directions,  and  that  their  actual 
size  is  so  small  compared  with  the  volume  of  the  vessel 
that  they  do  not  in  any  way  influence  each  other. 

108.  Boyle's  Law.     This  states  that,  if  the  temperature 
is  kept  unchanged,  but  the  density  of  the  gas  altered,  the 
pressure  is  always  proportional    to  the  density.      If  p  is 
pressure,  p  density,  v.  volume,  m  mass,  Boyle's  law  may  be 
written :  at  constant  temperature, 

p  =  k  p,     or    -p  v  =  k  m    .     .     .     .     (18) 

k  is  a  constant  for  a  given  gas  and  definite  temperature. 

By  most  elaborate  experiments  at  high  pressures  it  is 
found  that  Boyle's  law  is  not  rigidly  true,  but  that  the 
ratio  p  I  p  slightly  increases  for  enormous  pressures. 

If  the  volume  of  a  given  mass  of  gas  is  known  at  any 
pressure,  the  pressure  which  corresponds  to  any  other 
volume  may  be  calculated  if  the  temperature  has  not 
changed.  For  pv  =  pi  Vi.  An  apparatus  called  a  "  closed  " 
manometer  has  been  devised  on  this  principle  to  measure 
pressures.  A  strong  glass  tube  containing  air  is  fastened 
into  an  iron  box  full  of  mercury.  This  box  is  connected 


136 


THEORY   OF  PHYSICS. 


[CH.  IV 


with  the  space  where  the  pressure  is  to  be  determined. 

By  a  preliminary  experiment  the  volume,  v,  of  the  air 
in  the  tube  is  measured  when  the  pres- 
sure on  it  is  that  due  to  the  atmosphere ; 
and  when  the  mercury  is  subjected  to 
the  unknown  pressure,  pi}  let  the  air 
have  its  volume  changed  to  v^-  Hence, 
if  the  temperature  has  not  changed, 

p±  v\  =  v  X  atmospheric  pressure, 

and  so  p1  may  be  calculated. 

109.  Elasticity  of  a  Gas.  The  only 
coefficient  of  elasticity  for  a  gas  is  the 
one  corresponding  to  a  change  in  vol- 
ume ;  and  it  may  in  certain  cases  be 
easily  calculated.  For,  the  stress  is  the 
increase  in  pressure;  and  the  strain  is 
the  ratio  of  the  decrease  in  volume  to  the  original  volume, 
the  changes  being  very  small.  That  is,  if  p\,  vi,  are  the 
original  pressure  and  volume,  and  p2,  v2,  the  resulting 
ones  due  to  the  change, 

the  coefficient  of  elasticity  =  P*  "" Pl  =  Vl  ^  ~  Pl)   (19) 


FIG.  101. 


If  the  compression  takes  place  so  slowly  that  there  is 
no  change  in  temperature,  Boyle's  law  may  be  applied : 
ply  Vi  =  p2,  v%.  Hence 


P*  Oi  -  P«)  = 


(20) 


But,  since  the  change  in  the  pressure  has  been  pre- 
supposed very  small,  it  may  be  stated  that  the  coefficient 
of  elasticity  of  a  gas  at  constant  temperature  numerically 
equals  the  pressure  of  the  gas  at  that  instant. 

If  the  compression  takes  place  rapidly,  there  is  always 
an  increase  in  temperature ;  and  so  Boyle's  law  cannot 


110]     PROPERTIES  OF  SIZE  AND  SHAPE  OF  MATTER       137 

be  applied.  However,  if  the  gas  is  compressed  extremely 
rapidly,  so  that  there  is  no  time  for  it  to  give  out  heat 
to  the  surrounding  bodies,  it  may  be  proved  that  the 
coefficient  of  elasticity  of  the  gas  is  a  certain  constant 
times  the  pressure,  7  p,  where  7  has  the  value  for  air, 
hydrogen,  oxygen,  and  many  other  gases  of  about  1.40. 
The  nature  of  7  will  be  explained  later,  under  the  subject 
of  "  Specific  Heat  "  (Art.  182). 

110.  Air-pumps.  Just  as  water-pumps  are  used  to  draw 
water  out  of  one  vessel  and  pour  it  into  another,  so  pumps 
may  be  constructed  to  exhaust  a  gas  from  a  vessel  which 
contains  it.  Such  pumps  are  called  air-pumps,  and  are  of 
three  general  types. 

1.  Mechanical  pump.  This  is  in  principle  identical 
with  the  ordinary  liquid  lifting-pump,  with  three  valves, 
A  at  the  bottom  of  the  cylinder, 
C  at  the  top,  and  B  in  the  piston. 
The  vessel  from  which  the  gas  is 
to  be  exhausted  is  connected  in 
some  way  with  the  pipe  leading 
out  of  the  bottom  of  the  cylinder. 
As  the  piston  is  raised,  the  valve 
B  in  it  is  closed;  the  gas  above 
it  is  forced  out  into  the  air 
through  the  valve  G\  and,  the 


i 


valve  A    being  opened,  the  gas  FIG.  102. 

from  the  vessel  expands  into  the 

cylinder  below  the  piston.  If  the  piston  is  now  forced 
down,  the  valves  A  and  C  are  closed;  and,  the  valve  B 
in  the  piston  being  opened,  the  gas  below  the  piston 
passes  through  into  the  space  above.  So,  as  the  pro- 
cess continues,  more  and  more  of  the  gas  is  exhausted. 
The  three  valves  will  not  open  and  close  of  them- 
selves, as  they  would  in  a  water-pump,  but  are  made  to 
do  so  by  automatic  mechanisms  connected  with  the 
piston. 


138 


THEORY  OF  PHYSICS 


[CH.  IV 


2.  Sprengel  pump.     The  action   of  this  pump  consists 
in  having  drops  of  mercury  (or  some  other  liquid)  so  fall 

as  to  trap  the  gas  between  them,  and 
thus  carry  it  away.  There  is  an  elon- 
gated glass  bulb,  into  the  upper  end  of 
which  is  joined  a  tube  provided  with  a 
stop-cock  and  funnel  so  that  the  mercury 
may  be  thus  poured  in;  into  the  lower 
end  of  the  bulb  is  joined  a  glass  tube  of 
narrow  bore  and  at  least  80  cm.  long ; 
into  the  side  of  the  bulb  is  joined  a  con- 
nection with  the  space  to  be  exhausted. 
The  tubes  at  the  top  and  bottom  of  the 
bulb  are  so  arranged  that,  as  the  mercury- 
drops  break  off  and  fall,  they  hit  the 
opening  of  the  lower  tube  and  pass  down 
it  in  the  form  of  short  cylinders.  The 
space  between  these  cylinders  thus  formed 
is  occupied  by  small  amounts  of  the  gas, 
drawn  in  from  the  connected  vessel ;  and 
so  these  drops  act  like  a  succession  of  small  pistons  forcing 
out  the  gas.  The  lower  end  of  the  long  tube  may  dip  into 
a  basin  of  mercury  ;  and  the  gas  will  bubble  out  at  the 
surface.  As  the  exhaustion  continues,  the  mercury  will 
rise  in  the  long  tube,  and  will  finally  stand  at  the  baro- 
metric height  when  the  vacuum  is  as  complete  as  it  can  be 
made.  » 

3.  Geissler-Toepler   pump.     In    this   pump   there   is   a 
large  bulb  to  which  are  joined  two  tubes,  —  one  at  the  top, 
the  other  at  the  bottom.     The  lower  one  is  at  least  80  cm. 
long,  and  is  connected  at  its  lower  end  to  a  large  vessel  of 
mercury  by  means  of  a  long  rubber  tube.     The  upper  tube 
is  bent  over  into  a  vertical  direction  downward,  and  dips 
into  a  basin  of  mercury.     Around  the  large  bulb  there  is 
a  glass-tube  branch  connecting  the  upper  and  lower  tubes 
just  as  they  leave  the  bulb  ;  and  into  this  branch  is  joined 


FIG.  103. 


110]     PROPERTIES  OF  SIZE  AND  SHAPE  OF  MATTER      139 

a  long  vertical  tube  leading  to  the  vessel  which  is  to  be 
exhausted. 

If  the  large  vessel  of  mercury  is  now  raised,  as  it  can 
be  owing  to  the  flexible  rubber  tubing,  the  mercury  will 
rise  into  the  bulb  and  the  connecting  tubes,  shutting  off 
connection  with  the  vessel  to  be  exhausted,  and  will  drive 


FIG.  104. 


FIG.  105. 


out  all  the  gas  in  the  bulb  through  the  tube  in  the  top,  so 
that  it  will  bubble  out  through  the  mercury  in  the  basin 
at  its  end.  If,  now,  the  movable  vessel  of  mercury  is  low- 
ered, no  air  can  enter  through  the  tube  at  the  top  of  the 
bulb,  because  it  is  "sealed"  by  the  mercury  in  the  basin, 
which  will  rise  in  the  tube ;  but,  as  soon  as  the  mercury 
falls  below  the  opening  to  the  long  vertical  tube,  the  gas 
in  the  vessel  to  be  exhausted  will  expand  and  fill  the  bulb 


140 


THEORY  OF  PHYSICS 


[CH.  IV 


and  the  connecting  tubes.  When  the  movable  vessel  of 
mercury  is  again  raised,  it  drives  out  the  gas  in  the  bulb ; 
and,  as  the  process  continues,  the  exhaustion  of  the  vessel 
rapidly  proceeds.  The  tube  leading  from  the  top  of  the 
bulb  around  to  the  basin  of  mercury  must  be  at  least 
80  cm.  high  ;  and  the  long  vertical  tube  leading  to  the 
vessel  to  be  exhausted  must  be  still  longer. 

111.  Compression  Pumps.  One  often  desires  to  compress 
&>  gas  in  a  tube  or  cylinder,  and  for  this  purpose  so-called 
compression  pumps  are  used.  They  are,  in  principle,  noth- 
ing but  mechanical  air-pumps  with  their  valves  arranged 
so  as  to  open  in  the  opposite  directions.  A  simple  form  of 
cornpression-pump  is  the  ordinary  bicycle  pump,  a  section 
of  which  is  shown.  (See  Fig.  105.)  Around  the  inner  tube, 
A,  is  a  leather  "  washer,"  which  allows  air  to  pass  by  it  when 
the  outer  tube  is  drawn  away  from  the  tire.  The  outer  tube 
thus  becomes  filled  with  air ;  and,  if  it  is  now  pushed  in, 
the  washer  does  not  allow  the  air  to  escape  around  it,  and  so 
the  air  is  driven  into  the  tire  through  the  tube  A.  There 
is,  of  course,  an  opening,  B,  in  the  outer  tube,  joining  it  to 
the  open  air  so  as  to  allow  the  air  to  enter  at  each  stroke 
of  the  pump. 

TABLE  I 

DENSITIES 
SOLIDS 


Aluminum 

2.58 

Iron     

7.86 

Brass           (about) 

84 

11.3 

Copper     .... 
Diamond. 

8.92 
352 

Platinum       .     .     . 
Silver       .... 

21.50 
10.53 

Glass   common 

2.6 

Tin      

7.29 

<(      heavy  flint  . 

3.7 

Zinc     

7.15 

Ice  at  0°  C.      .     . 

0.9167 

Ill]     PROPERTIES  OF  SIZE  AND  SHAPE  OF  MATTER       141 

LIQUIDS 


Alcohol  at  20°  C.  . 
Ethyl  ether  at  0°C. 

0.789 
0.735 

Mercury  .... 
Water  at  4°  C.  .     . 

13.55 
1.000 

Water  at  other  temperatures,  see  below. 

GASES  AT  0°  C.  AND  76  CM.  of  MERCURY 


Air,  dry  .     .     . 

0.001293 

"Helium".     . 

0.00021 

11  Argon7'     .     . 

0.00170 

Hydrogen     .     . 

0.0000895 

Carbon    dioxide 

0.001965 

Nitrogen       .     . 

0.001254 

Chlorine  .     .     . 

0.00317 

Oxygen    .'  .     . 

0.001429 

WATER  AT  DIFFERENT  TEMPERATURES 


0°  C.  .  .  . 

0.999878 

16°  C.   .  . 

0.999004 

1°   ... 

0.999933 

17°    .  . 

0.998839 

2°    .  .  . 

0.999972 

18°     .  . 

0.998663 

3°    .'  .  . 

0.999993 

19°     .  . 

0.998475 

4°    ... 

1.000000 

20°    .  . 

0.998272 

5°    ... 

0.999992 

21° 

0.998065 

6°    ... 

0.999969 

22° 

0.997849 

7°    ... 

0.999933 

23°    .  . 

0.997623 

8°    ... 

0.999882 

24°    .  . 

0.997386 

9°    ... 

0.999819 

25°    .  . 

0.997140 

10°    ... 

0.999739 

26°     .  . 

0.99686 

11°    ... 

0.999650 

27°    .  . 

0.99659 

12°    ... 

0.999544 

28°     .  . 

0.99632 

13°    ... 

0.999430 

29°     .  . 

0.99600 

14°   ...  .  . 

0.999297 

30°    .  . 

0.99577 

15°    ... 

0.999154 

31°     .  . 

0.99547 

BOOK  .II 

SOUND 


BOOK   II 

SOUND 


INTRODUCTION 

To  any  one  with  the  sense  of  hearing,  the  name  "  sound  " 
conveys  a  definite  idea  of  a  particular  sensation.  In  Phys- 
ics the  theory  of  sound  is  the  study  of  the  nature  of  this 
sensation  and  of  the  exact  conditions  under  which  it  is 
produced. 

112.  It  is  a  matter  of  every-day  experience  that  sounds 
are  always  caused  by  vibrating  bodies  ;  e.  g.,  a  piano-string, 
the  air  in  an  organ-pipe,  an  explosion  or  sudden  blow.  It  is 
also  probably  well  known  to  every  one  that,  if  the  vibrating, 
i.  e.  "  sounding,"  body  is  far  away,  some  time  elapses  be- 
tween the  vibration  and  the  perception  of  the  sensation. 
Thus  it  is  noticed  that  there  is  an  interval  of  time  between 
the  flash  of  a  pistol  and  the  sound  of  its  report ;  another 
illustration  is  the  interval  between  the  flash  of  lightning 
and  the  consequent  thunder.  It  may  also  be  shown  that, 
if  there  is  no  material  medium  between  the  sounding  body 
and  the  ear,  no  sound  is  heard.  For,  if  a  bell  is  rung  in- 
side a  vacuum,  the  ear  hears  no  sound.  This  proves  that 
the  presence  of  matter  is  essential  for  the  propagation  of 
the  disturbance  that  produces  sound. 

Further,  it  is  important  to  emphasize  the  fact  that  sound 
is  a  sensation  perceived  in  the  ear.  The  sound  is  not  emit- 
ted by  the  vibrating  body,  but  is  produced  by  it. 


CHAPTER  I 

VIBRATIONS 

113.  Detection  of  Vibrations.  That  sounding  bodies  of  all 
kinds  are  in  vibration  admits  of  immediate  proof.  Gener- 
ally, the  sense  of  touch  is  sufficient  to  convince  one  of  the 
fact;  for  the  vibration  may  be  felt.  This  is  true  in  the 
case  of  solid  bodies,  such  as  a  stretched  string,  a  tuning- 
fork,  or  a  bell.  When  columns  of  air  in  organ-pipes  are 
causing  sounds,  the  air  may  be  easily  proved  to  be  vibrat- 
ing by  placing  at  different  points  in  the  pipe  mem- 
branes carrying  some  light  powder.  The  powder  will  in 
certain  places  be  violently  shaken.  Similar  means  may 
be  used  to  detect  the  vibration  in  the  case  of  sounding 
columns  of  a  liquid. 

Nature  of  the  Vibrations.  The  exact  nature  of  the  vibra- 
tions may  also  be  studied  by  suitable  methods.  One  of  the 


FIG.  106. 


simplest  is  to  place  a  stiff  bristle  upon  the  body,  and  then 
while  it  is  vibrating  to  draw  under  it  a  piece  of  smoked 
glass,  so  that  the  bristle  traces  on  the  glass  a  path  which 
corresponds  to  the  vibration  of  the  body  and  the  motion 


114]  VIBRATIONS  147 

of  the  glass.     Such  a  method  applied  to  a  tuning-fork  is 
shown  in  the  figure. 

Another  method  is  to  study  the  effect  on  the  air  in  the 
neighborhood  of  the  sounding  body  by  placing  near  the 
body  a  stretched  membrane  to  which  is  attached  a  bristle  ; 
and  this  can  record  the  vibration  of  the  air  on  a  smoked 
plate. 

(By  having  the  glass  plate  attached  to  a  second  vibrating 
body,  the  combined  motion  of  the  two  bodies  may  be  regis- 
tered and  studied.  This  may  also  be  done  by  suitable 
optical  methods.) 

In  this  way  it  is  proved  that  there  are  two  classes  of 
sounding  bodies.  One  class  makes  a  series  of  periodic  vi- 
brations, the  other  does  not.  A  periodic  vibration  is  a 
motion  such  that  at  regular  intervals  of  time  the  identical 
process  is  repeated.  This  interval  is  called  the  "  period  "  of 
the  vibration  ;  and  the  number  of  complete  vibrations  in  one 
second  is  called  the  "  frequency."  If  T  is  the  period,  and  n 
the  frequency,  it  is  obvious  that  T  =  1  /  n.  Illustrations 
of  this  motion  are  the  vibrations  of  tuning-forks,  piano- 
strings,  and  all  musical  •  instruments. 

Illustrations  of  non-periodic  vibrations  are  the  motions 
produced  by  a  series  of  sudden  blows,  like  a  hammer  on  an 
anvil,  a  carriage  rolling  over  a  rough  pavement,  the  tearing 
of  a  piece  of  paper,  and  all  so-called  noises.  It  may  be 
shown  that  noises  are  due  to  a  combination  of  a  great  num- 
ber of  periodic  motions  of  nearly  the  same  periods,  but 
which  last  only  an  extremely  short  time. 

The  "  amplitude  "  of  any  vibration  is  the  extent  of  the 
motion  between  the  extreme  limits ;  and  it  is  evident  that 
the  greater  the  amplitude  is,  so  much  greater  is  the  energy 
of  the  vibration. 

114.  Periodic  Vibrations.  The  simplest  type  of  periodic 
vibration  is  harmonic  motion  (see  Art.  21)  ;  but  illus- 
trations of  this  are  by  no  means  common  among  musical 
instruments.  A  tuning-fork  or  a  stretched  string  may  be 


148  THEORY  OF  PHYSICS  [CH.  I 

made  to  vibrate  with  harmonic  motion ;  but  it  is  not  easy. 
In  general,  the  motion  is  much  more  complicated.  It  may 
be  proved,  however,  that  any  periodic  motion,  produced  in 
any  way,  is  equivalent  in  every  respect  to  the  combination 
of  a  number  of  harmonic  motions  of  suitable  periods  and 
amplitudes.  If  the  general  periodic  motion  has  the  period 
T,  the  harmonic  components  must  have  the  periods  T, 
T/2,  T/3,  T/4,  etc.  The  component  with  the  period  T 
is  called  the  "  fundamental "  vibration ;  the  others  are 
sometimes  called  the  "  upper  partial "  vibrations.  (This 
statement  is  known  as  Fourier's  Theorem ;  and  it  admits 
of  rigid  mathematical  demonstration.) 


FIG.  107. 

This  question  of  the  combination  of  harmonic  vibrations 
so  as  to  produce  any  periodic  motion  may  be  best  illustrated 
graphically.  Layoff  along  a  horizontal  line  points  cor- 
responding to  equal  intervals  of  time ;  and  at  each  point  of 
this  line  erect  a  perpendicular  line  equal  in  length  to  the 
displacement  at  that  instant  of  the  vibrating  body  from  its 
position  of  equilibrium.  If  the  motion  is  harmonic,  the 
curve  which  is  the  locus  of  the  ends  of  these  perpendicular 
lines  will  be  as  shown.  (This  curve  is  called  a  "  sine-curve.") 
Any  other  motion  will  have  its  corresponding  curve ;  and  the 
combination  of  various  vibrations  may  be  shown  by  sim- 
ply adding  algebraically  their  curves.  Thus,  if  there  are 
two  harmonic  vibrations,  whose  periods  are  T  and  T/2 


115] 


VIBRATIONS 


149 


their  curves  are  as  shown ;  and  their  combination  is  given 
by  simply  adding  the  displacements  which  correspond  to 
the  same  instants  of  time.  The  resulting  curve  will,  of 
course,  depend  upon  how  much  one  vibration  lags  behind 
the  other ;  that  is,  upon  whether  they  begin  together  or  at 
different  instants.  The  curve  shown  corresponds  to  the 
case  when  the  two  vibrations  begin  at  the  same  time ;  and 
the  resulting  motion  is  seen  to  be  periodic  with  the  period 
T.  In  a  similar  manner  the  most  general  possible  combi- 
nation may  be  treated. 


FIG.  108. 


The  exact  nature  of  the  vibrations  of  musical  instru- 
ments, and  the  discussion  of  the  individual  properties  of 
each  instrument,  will  be  given  later,  in  Chapter  V. 

The  essential  characteristics  of  any  periodic  vibration 
are,  then,  (1)  frequency ;  (2)  amplitude ;  (3)  complexity, 
or  nature  of  component  harmonic  vibrations. 

115.  Measurement  of  the  Period  of  Vibration.  It  is  not  dif- 
ficult to  measure  the  period  of  vibration  of  a  vibrating 
body,  or  the  reciprocal  of  this  quantity,  the  frequency ; 
and  various  methods  are  taught  in  laboratories.  One 
method  is  to  attach  a  bristle  to  the  vibrating  body, 
and  to  move  under  the  bristle  a  piece  of  smoked  glass 
at  a  known  velocity.  (E.  g.,  fasten  the  glass  to  a  pen- 
dulum and  measure  its  period  of  vibration,  or  allow 
the  glass  to  fall  vertically  downward  toward  the  earth.) 
Then  count  the  number  of  vibrations  registered  in  a 
measured  space  on  the  glass,  and  calculate  from  the 
known  velocity  of  the  glass  what  portion  of  a  second 


150 


THEORY  OF  PHYSICS 


[CH.  I 


has  been  taken  in  moving  this  distance;  the  ratio  of 
this  time  to  the  number  of  vibrations  counted  is  the 
period  of  the  vibrating  body.  Another  method  is  to  com- 
pare the  period  of  the  body  with  that  of  a  simple  pen- 
dulum directly  by  means  of  "  coincidences."  Again,  some 
instruments  are  so  constructed  as  to  register  automatically 
the  number  of  vibrations.  The 
best  instrument  of  this  type  is  a 
so-called  "  siren."  This  consists 
of  a  circular  disc  which  contains 
several  series  of  openings  ar- 
ranged in  concentric  circles,  and 
which  can  rotate  in  its  own 
plane  around  an  axis  through  its 
centre.  These  openings  do  nofc 
pass  vertically  through  the  disc, 
but  each  one  slants  tangentially 
slightly ;  so  that,  when  a  blast 
of  air  is  blown  through  it,  there 
is  a  pressure  produced  on  the 
walls  of  the  passage  tending  to 
make  the  disc  revolve.  If,  then, 
a  blast  is  forced  up  through  a 
tube  directed  so  as  to  strike  a 
series  of  openings,  there  will  be  a  succession  of  impulses 
given  to  the  air  above  the  disc ;  because,  being  set  in  mo- 
tion by  the  blast,  the  disc  will  revolve,  and  so  the  openings 
will  come  in  turn  over  the  mouth  of  the  tube  conveying  the 
air.  The  number  of  impulses  per  second  given  the  air  equals 
the  number  of  openings  multiplied  by  the  number  of  revolu- 
tions per  second  of  the  disc.  By  altering  the  number  of 
openings  or  the  pressure  of  the  air-bfest,  the  number  of 
vibrations  can  be  changed;  and  it  is  perfectly  simple  to 
have  the  entire  number  of  revolutions  of  the  disc  registered 
mechanically  by  a  screw  and  dial.  So  the  frequency  of 
the  vibration  can  be  at  once  deduced.  Another  method 
will  be  described  later  (see  Art.  126). 


CHAPTER  II 
SOUND  WAVES 

As  already  stated,  an  interval  of  time  elapses  between 
the  vibration  of  the  sounding  body  and  the  perception  of 
sound  in  our  ears  ;  and  the  presence  of  ordinary  matter  is 
necessary  to  carry  the  effect  from  the  body  to  the  ear.  In 
this  chapter  the  method  of  the  propagation  of  this  effect 
will  be  discussed. 

116.  Existence  of  Waves.     As  the  sounding  body  vibrates, 
it  gives  a  series  of  impulses  to  whatever  matter  is  touching 
it ;  and  so  a  disturbance  is  sent  out  into  the  surrounding 
medium.     Thus,  a  tuning-fork  vibrating  in  air  or  in  water 
sends  out  waves  ;  if  it  touches  a  solid  like  an  iron  pipe  or 
a  wooden  table,  it  sends  waves  out  in  them  also ;  because, 
by  definition,  a  wave  is  simply  the  advance  of  a  disturb- 
ance into  a  medium.     See  Articles  74  et  scq. 

117.  Nature  of  Sound  Waves.     Since  all  sounding  bodies 
are  making  vibrations,  to  and  fro,  they  produce  simply  a 
"  push  and  pull "  in  the  surrounding  medium.     Therefore, 
the  portions  of  the  medium  move  backward  and  forward 
in  the  direction  of  the  advance  of  the  wave ;  that  is,  the 
waves  which   produce  sound   are   longitudinal  or   "com- 
pressional." 

Consider  more  in  detail  the  simplest  case,  a  tuning-fork 
vibrating  in  the  air.  Each  time  the  prongs  move  out  they 
produce  a  condensation  of  the  air  immediately  in  front; 
and  owing  to  the  elasticity  of  the  air  this  condensation 
spreads  out  into  the  air.  Then,  as  the  prongs  return,  they 
produce  a  rarefaction,  which  also  extends  out  into  the  air. 


152 


THEORY  OF  PHYSICS 


[CH.  II 


So,  as  the  vibrations  continue,  there  is  a  succession  of  con- 
densations and  rarefactions  sent  out.  The  individual  par- 
ticles of  the  air  do  not,  of  course,  move  onward  with  the 
wave,  but  vibrate  backward  and  forward  in  the  direction 
of  the  advance  of  the  waves. 

A  very  good  illustration  of  this  particular  kind  of  wave 
is  given  by  a  model  consisting  of  a  series  of  heavy  spheres 
suspended  in  line  at 
regujar  intervals  by 
long  threads  and  sepa- 
rated by  light  spiral 
springs.  If  the  first 
sphere  is  moved  toward 
the  right,  it  compresses 
the  first  spring ;  then 
the  second  sphere  is 
moved;  this  compresses 
the  second  spring,  etc. 
The  compressional  wave 
advances  along  the  row  FIG.  110. 

of  spheres  with  a  veloc- 
ity depending  upon  the  mass  of  the  spheres  and  the  elas- 
ticity of  the  springs.  Similarly,  if  the  first  sphere  is  moved 
toward  the  left,  the  spring  is  stretched ;  and  so  a  rarefac- 
tion advances  along  the  line,  with  the  same  velocity  as  the 
compressional  wave  did.  If  the  first  sphere  is  made  to 
vibrate  periodically,  there  will  be  a  succession  of  condensa- 
tions and  rarefactions  sent  out.  The  individual  spheres 
make  periodic  vibrations,  as  is  evident,  but  do  not  change 
their  positions  of  equilibrium. 

If  the  waves  are  in  any  other  medium  than  air,  they 
also  consist  in  the  advance  of  condensations  and  rare- 
factions. 

118.  Characteristics  of  the  Waves.  When  a  train  of  waves 
advances  into  any  medium,  it  has  certain  fundamental 
characteristics.  The  wave-length,  X,  is  the  distance  from 


119]  SOUND  WAVES  153 

condensation  to  condensation,  or  from  any  particle  to 
the  next  particle  in  the  direction  of  the  advance  of 
waves,  which  is  in  identically  the  same  position  and 
condition  relative  to  its  vibration.  The  "  frequency  "  or 
"wave-number,"  n,.  is  the  number  of  condensations  (or 
rarefactions)  sent  out  in  one  second,  or,  what  is  the  same 
thing,  the  number  of  condensations  (or  rarefactions)  which 
pass  by  a  fixed  point  in  the  medium  during  one  second. 
The  velocity,  v,  is  the  distance  the  waves  advance  in  one 
second.  Hence  v  =  n  X.  (This  velocity  has,  of  course, 
no  connection  with  the  speed  of  the  individual  particles  of 
the  medium  ;  for  the  latter  is  changing  each  instant,  and 
depends  upon  the  intensity  of  the  vibrations  of  the  sound- 
ing body.)  The  "amplitude"  is  the  length  of  the  path 
through  which  each  particle  of  the  medium  vibrates.  The 
waves  carry  away  energy  from  the  sounding  body;  and 
the  intensity  of  the  waves  (see  Art.  74)  is  inversely  pro- 
portional to  the  square  of  the  distance  from  the  source 
and  directly  proportional  to  the  square  of  the  amplitude. 
(This  statement  refers  to  the  intensity  of  the  waves,  not  to 
the  intensity  of  the  sound  sensation.) 

119.  Velocity,  As  is  evident,  the  velocity  of  the  waves 
depends  upon  the  elasticity  and  the  inertia  of  the  medium  ; 
and  it  may  be  proved  that  for  any  elastic  wave  the  value  of 
the  velocity  in  an  isotropic  medium  is 


—  , 
p 

where  E  is  the  particular  coefficient  of  elasticity  corre- 
sponding to  the  strain  characteristic  of  the  waves,  and  p  is 
the  density.  It  is  seen,  then,  that  the  velocity  is  entirely 
independent  of  the  length  or  the  amplitude  of  the  wave, 
since  E  and  p  are  in  general  constants.  Therefore,  since 
v  =  n  X,  if  the  frequency  is  small,  the  waves  are  long  ;  and 
conversely.  In  sound  waves,  the  strain  is  one  of  change 
of  volume;  and  so  E  is  the  bulk-modulus  (see  Art.  81). 


154  THEORY  OF  PHYSICS  [CH.  II 

Further,  the  compressions  and  rarefactions  are  very  rapid  ; 
and  so  no  heat  can  enter  or  escape. 

For  a  gas,  then,  the  coefficient  of  elasticity  is  y  p  (see 
Art.  109),  where  p  is  the  pressure  and  7  is  a  constant  for 
any  one  gas.  Thus 


because,  as  will  be  shown  later  (Heat,  Art.  178)  for  a 
gas  p  =  R  p  T,  where  E  is  a  constant  for  any  one  gas,  and 
T=273  +  t°  on  the  centigrade  scale  of  temperature.  The 
velocity  in  a  gas  is  thus  independent  of  wave-length,  of 
amplitude,  and  of  density  of  the  medium,  but  varies  as  the 
square  root  of  273  -f-  1°  0.  As  will  be  shown  later,  v  can 
be  easily  measured  for  any  gas,  so  can  R  and  t  ;  conse- 
quently 7  for  that  gas  may  be  thus  determined  ;  and  this 
in  fact  is  the  method  generally  adopted. 

120.  Composition  of  Waves.  If  the  sounding  body  is 
making  harmonic  vibrations,  it  will  send  out  what  may  be 
called  a  harmonic  train  of  waves.  These  waves  will  be 
spherical  ;  but,  at  some  distance  from  the  vibrating  body, 
they  may  be  regarded  as  practically  plane.  (See  Art. 
75.)  Each  particle  of  the  medium  is  making  harmonic 


FIG.  ill. 


vibrations  through  a  position  of  equilibrium;  and  so  the 
motion  is  handed  on.  This  may  be  represented  graphically 
in  this  way :  Draw  a  line  in  the  direction  of  the  advance 
of  the  waves ;  lay  off  on  it  points  to  represent  the  positions 
of  equilibrium  of  various  particles  of  the  medium;  erect 


121]  SOUND  WAVES  155 

at  these  points  perpendicular  lines  whose  lengths  equal 
the  displacements  of  those  particles  at  any  instant  from 
their  positions  of  equilibrium ;  the  locus  of  the  extremities 
of  these  lines  may  be  called  the  "  wave-form."  If  the 
vibration  is  harmonic,  the  wave-form  will  be  a  sine-curve, 
as  shown.  In  a  sound  wave,  the  particles  are  moving  in 
the  direction  of  the  advance  of  the  wave,  not  across  the 
direction ;  but  for  the  sake  of  clearness  in  the  diagram  the 
displacements  are  represented  as  if  they  were  transverse. 

At  any  instant  later,  the  wave-form  will  be  simply  this 
first  form  moved  forward  in  the  direction  of  advance  of  the 
waves  ;  and  so  the  entire  phenomenon  of  the  train  of 
waves  may  be  considered  graphically  as  the  motion  for- 
ward of  this  wave-form  with  the  velocity  of  the  waves. 

If  the  sounding  body  is  making  periodic  vibrations, 
which  are  not  harmonic,  these  may  be  resolved,  as  ex- 
pained  in  Article  114,  into  a  series  of  harmonic  vibrations 
of  periods,  T,  T / 2,  T/ 3,  etc.;  and  each  of  these  partial 
harmonic  vibrations  may  be  considered  as  sending  out  a 
harmonic  train  of  waves.  So  the  resulting  train  of  waves, 
due  to  any  periodic  vibration,  may  be  regarded  as  the  sum 
of  a  number  of  harmonic  trains.  The  form  of  the  waves 
will,  of  course,  depend  upon  the  amplitudes  of  these 
separate  waves.  Consequently,  several  trains  of  waves 
may  have  the  same  period  and  amplitude,  but  different 
"  forms." 

Similarly,  if  two  or  more  sounding  bodies  are  sending 
out  waves  into  the  same  medium,  the  resultant  displace- 
ment at  any  point  will  be  the  geometrical  sum  of  the 
displacements  which  each  wave  separately  would  have 
produced,  provided  only  that  these  individual  displacements 
are  small  in  comparison  with  the  length  of  the  waves. 
This  is  perfectly  analogous  to  the  combined  action  of  two 
overlapping  water-waves  whose  crests  are  not  too  high. 

121.  It  is  thus  evident  that  there  are  three  essential 
properties  of  a  given  train  of  waves  :  first,  its  wave-number 


156  *  THEORY  OF  PHYSICS  [CH.  II 

or  frequency ;  second,  its  amplitude ;  third,  its  form.  If  the 
frequency  is  known,  so  is  the  wave-length,  because  in  a  given 
medium  the  velocity  is  the  same  for  waves  of  all  wave- 
lengths ;  and_£  =  n  X.  Therefore,  for  a  given  value  of  n, 
there  is  a  definite  value  of  X  in  any  one  medium. 

These  three  characteristics  of  a  train  of  waves  corre- 
spond perfectly  to  the  three  characteristics  of  a  vibration : 
frequency,  amplitude,  and  complexity. 


CHAPTER    III 

SOUND   SENSATION 

122.  Perception  of  Sound.  These  compressional  waves,  sent 
out  by  the  vibrating  source,  spread  through  the  surround- 
ing matter ;  and,  if  a  being  with  the  sense  of  hearing  is  in 
the  neighborhood,  the  waves  will,  in  general,  produce  in  his 
ear  the  sensation  of  sound.  All  compressional  waves  do 
not  produce  sounds ;  because  it  may  be  proved  by  experi- 
ment that,  if  the  frequency  is  too  small  or  too  great,  no 
sensation  is  perceived.  These  limits  vary  greatly  for  dif- 
ferent individuals  ;  but  roughly  it  may  be  said  that,  if  the 
frequency  of  the  waves  is  between  20  and  40,000,  sounds 
will  be  produced  in  the  human  ear.  If  the  frequency  of 
the  vibrating  body  is  less  than  twenty  vibrations  per 
second,  it  is  doubtful  if  a  true  compressional  wave  is 
produced  in  a  fluid ;  because,  between  consecutive  vibra- 
tions, the  fluid  will  have  time  to  flow  around  the  vibrating 
body,  and  will  not  be  simply  compressed  by  it.  If  the 
frequency  is  greater  than  forty  thousand  (in  fact  even  if 
it  is  considerably  less)  the  waves,  although  they  may  pro- 
duce no  sound  in  the  ear,  may  be  traced,  and  their  laws 
studied,  by  means  of  what  is  known  as  a  "  sensitive  flame." 
If  any  illuminating  gas  is  allowed  to  escape  into  the  air 
from  a  vessel  which  holds  it  under  pressure,  it  will  form 
a  jet  as  it  comes  out ;  and,  if  this  is  lighted,  and  the 
pressure  so  regulated  that  the  jet  does  just  not  flicker,  its 
stability  is  exceedingly  unstable.  If  a  body,  even  some  dis- 
tance away,  makes  vibrations  of  a  great  frequency,  the  jet 
will  collapse,  owing  to  the  effect  of  the  very  short  waves 


158 


THEORY  OF  PHYSICS 


[CH.  Ill 


upon  the  surface  which  separates  the  jet  from  the  air. 
The  particularly  sensitive  point  of  the  jet  is  near  its  base, 
where  it  emerges  into  the  air. 

123.  The  Ear.    The  waves  enter  the 
ear  through  the  air,  and  necessarily 
set  in  vibration  the  "  drum  "  of  the 
ear,  which  closes  the  passage.     This 
membrane  is   connected   by  a  series 
of  bones  with  the  inner  cavity  of  the 
ear,  which  contains  a  liquid  substance. 
And  it  is  in  the  inner  ear,  within  the 
influence  of  this  liquid,  that  the  au- 
ditory nerves  end.     Consequently  any 
vibrations    of  the   ear-drum   produce 
corresponding    motions    around    the 
nerve-endings;    but  what   the   exact 
connection    is    between     these    two, 
i.  e.  the  cause  and  effect,  is  at  present 
unknown. 

124.  Musical  Notes,     The   simplest 
vibration  possible  for  a  sounding  body 
is,  as  stated  above,  a  harmonic  motion. 
It  is  a  fact  of  experiment,  too,  that 
the  simplest  sound  sensation  known 

FIG  112  to  us  is  that  produced  by  an  instru- 

ment which  vibrates  harmonically. 
Such  a  sound  is  called  a  "tone."  If  an  instrument  is 
vibrating  any  other  way,  it  emits  a  train  of  periodic  waves, 
which  produces  in  the  ear  a  more  complex  sensation.  But 
it  was  stated  that  any  such  train  of  periodic  waves  may  be 
considered  as  the  sum  of  a  number  of  harmonic  waves ;  and 
it  is  also  a  fact  of  experiment  that,  when  a  train  of  periodic 
waves  comes  into  the  ear,  the  sound  sensation  is  not  a 
single  one  ;  but,  on  the  contrary,  several  simple  tones  may 
be  distinguished,  and  the  sensation  is  thus  complex.  In 
fact  it  seems  to  be,a  general  law  that,  no  matter  how  many 


127]  SOUND   SENSATION  159 

trains  of  waves  enter  the  ear,  or  how  complicated  they 
are,  the  complex  sound  heard  may  be  mentally  resolved 
into  the  simple  tones  corresponding  to  the  harmonic  waves 
composing  the  waves.  This  statement  is  sometimes  called 
"  Ohm's  Law  of  Sound  Sensation." 

125.  Characteristics  of  Musical  Notes.    If  two  musical  notes 
are  compared,  the  sensations  differ  in  three  respects.     One 
may  be  shriller  or  "  higher "  than  the  other ;  one  may  be 
louder  or   more   "intense"  than   the  other;   and  finally, 
there  is  a  marked  difference  depending  upon  the  instru- 
ment which  produces  the  sound. 

126.  Pitch.    The  "  pitch  "  of  a  note  is  the  name  given  to 
that  property  which  marks  its  shrillness.     And  it  is  easily 
proved  by  experiment  that,  as  the  number  of  waves  per 
second  entering  the  ear  increases,  so  does  the  pitch  of  the 
sound  sensation  grow  higher.     If  the  distance  between  the 
ear  and  instrument  emitting  the  waves  is  not  changing, 
the  number  of  waves  entering  the  ear  equals  the  number 
of  waves  sent  off  by  the  instrument.    Two  vibrating  bodies, 
then,    under    these    conditions,    which    have    the    same 
frequency,  produce  sounds  of  the  same  pitch.     That  is,  the 
pitch  of  the  SQund  may  be  defined  numerically  as  being 
equal  to  the  frequency  of  the  vibration  or  the  frequency  of 
the  train  of  waves.     A  person  with  a  musical  education 
can  easily  distinguish  between  two  notes  whose  pitches 
are  extremely  close ;  and  so  the  frequency  of  any  vibrating 
body  may  often  be  determined  by  comparing  the  pitch  of 
the  sound  which  it  produces  with  that  produced  by  some 
instrument  whose  frequency  can  be  accurately  measured 
and  also  varied.     The  pitch  is  slowly  altered  until  its  note 
and  that  of  the  first  body  have  the  same  pitch,  that  is, 
are  in  "unison."     Such  an  instrument,  whose  frequency 
can  be  varied  continuously  through  a  wide  range,  and  can 
at  the  same  time  be  accurately  measured,  is  the  "siren." 

127.    If,  however,  the  vibrating  body  is  approaching  the 
ear  or  receding  from  it,  the  number  of  waves  which  enter 


160  THEORY   OF  PHYSICS  [CH.  Ill 

the  ear  in  one  second  no  longer  equals  the  number  emitted 
by  the  body.  There  is  a  greater  or  a  less  number  of  con- 
densations or  "crests"  in  a  given  space  in  the  interven- 
ing medium  than  there  would  be  if  the  body  were  not 
moving;  that  is,  the  wave-length  is  changed. 

Let  n  =  frequency  of  sounding  body. 

V  —  velocity  of  compressional  waves. 
v  —  velocity  of  body  toward  the  ear. 

Since  V  is  the  velocity  of  the  waves,  this  means  that  in 
one  second  a  crest  which  is  at  a  distance  V  from  the  ear 
advances  to  the  ear;  and  the  number  of  crests,  then,  in 
this  distance  Centers  the  ear  in  one  second.  Owing,  though, 
to  the  advance  of  the  sounding  body,  the  n  waves  or  crests 
which  it  emits  in  one  second  are  comprised  in  a  space 
V—  v ;  and  so  in  a  space  V  there  are  n'  crests,  where 
n' :  n  =  V:  V—v.  Hence  the  pitch  of  the  sound  heard  is 

nV 

n  = 


V-v 

Similarly,  if  the  sounding  body  is  receding  from  the  ear 
with,  the  velocity  v,  the  pitch  of  the  sound  heard  is 


n'  = 


This  explains  the  change  in  pitch  of  the  whistle  of  a  loco- 
motive when  it  approaches  or  recedes  from  an  observer. 

If  the  sounding  body  is  at  rest,  and  the  person  listening 
is  approaching  or  receding,  the  case  is  somewhat  different. 
Let  n  and  V  be  as  before,  and  let  v  be  the  velocity  of  the 
person  away  from  the  sounding  body.  There  are  n  crests 
in  a  distance  V ;  and  so,  as  the  person  moves  away  from 
the  source  of  the  waves  a  distance  v  in  one  second,  the 


129]  SOUND   SENSATION  161 

number  of   crests  which  overtake  and   pass  him  in  one 
second  is  nf,  where 

n':n=  V-v:  V. 

Hence  the  pitch  of  the  sound  is  heard 

n(V-v) 

—     77" ~     * 

Similarly,  if  the  person  is  approaching  the  sounding  body 
with  a  velocity  v,  the  pitch  of  the  sound  heard  is 


This  statement  about  the  change  in  pitch  of  a  sound  due 
to  the  alteration  in  position  of  the  sounding  body  and  the 
hearer  is  called  "  Db'ppler's  Principle  ;  "  and  it  may  obvi- 
ously be  extended  to  any  system  of  waves  in  any  medium. 

128.  Intensity.      The  loudness  or  intensity  of   a  sound 
may  be  shown  to  vary  with  the  intensity  of  the  waves  ; 
that  is,  it  varies  with  the  remoteness  of  the  sounding  body 
and  with  the  amplitude  of  the  waves.     But  one  is  not  a 
measure  of  the  other.     In  fact,  as  yet,  no  numerical  value 
can  be  given  the  intensity  of  a  sensation  ;  and  it  is  doubt- 
ful if  in  any  case  it  would  be  exactly  proportional  to  the 
energy  received  by  the  ear.     All  that  can  be  said  is  that 
the  intensity  of  any  sound  may  be  increased  by  increasing 
the  intensity  of  the  waves,  and  vice  versa. 

129.  Quality.     That  difference  which  may  exist  between 
sounds  of  the  same  pitch  and  comparative  intensity,  and 
which  enables  one  to  distinguish  between  them,  is  said  to 
be  a  difference  in  "  quality."     It  has  been  shown  by  care- 
ful experiments  to  depend  upon  the  "  form  "  of  the  waves, 
that  is,  upon  the  partial  vibrations  which  are  superimposed 
upon  the  fundamental.     Every  instrument  has  particular 
partials  which  it  produces;  and  they  and  their  relative 


162 


THEORY  OF  PHYSICS 


[CH.  Ill 


intensities  vary  with  different  instruments,  thus  producing 
differences  in  the  forms  of  the  waves.  Any  vibration  may 
be  analyzed  into  its  fundamental  and  upper  partials  by 
means  of  "  resonators,"  which  are  instruments  designed  to 
pick  out  from  any  complex  sound  its  component  parts. 
The  principle  made  use  of  is  a  most  general  mechanical 
one :  If  a  train  of  waves  of  a  certain  frequency  is  passing 
through  any  medium  in  which  is  a  body  which,  when  it 
vibrates,  has  the  same  frequency  as  the  wave,  then  this 
body  will  itself  ba  set  in  vibration  by  the  waves.  The 
body  will  begin  with  a  very  minute  motion  ;  but  the  suc- 


FIG.  113. 

cessive  impulses  given  by  the  waves  are  so  timed  as  to 
further  increase  the  motion ;  just  as,  by  pulling  at  the 
proper  instants,  a  man  may  set  a  heavy  bell  in  vibration. 
An  illustration  of  this  principle  is  given  when  two  tuning- 
forks  of  the  same  frequency  are  placed  near  each  other, 
and  one  is  set  in  vibration  by  means  of  some  blow.  If 
this  one's  motion  is  now  stopped,  it  will  be  noticed  that 
the  other  one  is  in  vibration.  The  commonest  form  of 
resonator  is  a  hollow  sphere  which  has  two  small  openings 
on  opposite  sides.  A  long  cylinder,  closed  at  one  end,  is 


132]  SOUND   SENSATION  163 

also  often  used.  The  air  inside  may  be  set  in  vibration  by 
blowing  over  the  mouth  of  the  opening ;  and  it  has  a  defi- 
nite frequency  depending  upon  its  size  and  shape.  If, 
now,  a  vibration  of  this  same  pitch  is  given  by  a  neigh- 
boring body,  the  air  in  the  resonator  will  be  set  in  "sympa- 
thetic "  vibration  ;  and  so  the  sound  heard  will  be  strength- 
ened. A  vibration  of  any  other  frequency  would  not  be 
thus  reinforced.  So,  by  having  a  series  of  resonators  of  dif- 
ferent frequencies,  any  complex  vibration  may  be  resolved 
into  its  components  ;  and  it  may  thus  be  proved  that  differ- 
ences in  quality  are  due  to  differences  in  the  number  and 
intensity  of  the  partials  present.  It  should  be  remembered 
that  if  the  fundamental  of  any  note  has  a  period  T,  its  upper 
partials  have  the  period  T/  2,  T /  3,  etc. ;  that  is,  if  the  fre- 
quency of  the  fundamental  is  n  (=  1  /  T),  the  frequencies 
of  the  partials  are  2  n,  3  n,  etc. 

130.  It  is  thus  seen  that,  corresponding  to  the  three 
physical  properties  of  a  train  of  waves,  its  wave-number, 
amplitude,  and  form,  there  are  three  characteristics  of  any 
musical  note,  —  its  pitch,  intensity,  and  quality. 

131.  Combinations  of  Notes.     If  two  or  more   trains  of 
waves  are  passing  through  the  same  medium,  the  resulting 
displacement  at  any  point  is,  of  course,  the  geometrical  sum 
of  the  individual  displacements  which  each  train  of  waves 
by  itself  would  have  produced.     If  two  trains  of  waves  so 
overlap  that  at  some  point  the  "  crest "  of  one  coincides  with 
the  "  hollow  "  of  the  other,  —  that  is,  if  the  displacements 
due  to  the  two  waves  are  in  opposite  directions  at  that 
point, —  there  will  be  no  motion  at  that  point  if  the  am- 
plitudes of  the  two  waves  are  equal.     But  if  two  crests 
coincide,  the  displacement  will  be  abnormally  great.     This 
phenomenon  of  the  overlapping   of   waves   is   sometimes 
called   "  interference ; "    and,  when   one   wave   neutralizes 
another  at  any  point,  the  interference  is  said  to  be  com- 
plete.    There  are  corresponding  properties  of  sensation. 

132.  Beats.     If  two  sounding  bodies,  e.  g.  two  tuning- 


164  THEORY  OF  PHYSICS  [CH.  Ill 

forks,  of  different  frequencies  are  vibrating  near  each  other, 
each  sends  out  a  train  of  waves  which  reaches  the  ear  of  an 
observer  near  by.  But  the  wave-form  of  the  combined 
motion  contains  places  of  no  motion  if  the  amplitudes  are 
the  same,  and  places  of  abnormal  motion,  as  is  shown  in  the 
diagram,  which  represents  the  combination  of  two  harmonic 
trains  of  waves  whose  frequencies  are  in  the  ratio  of  two  to 
three.  In  general,  since  one  train  of  waves  has  more  "  crests  " 


FIG.  114. 


in  a  given  space  than  the  other,  they  will  completely  in- 
terfere at  certain  points,  the  number  of  which  in  a  distance 
equal  to  the  velocity  of  the  waves,  equals  the  difference  of 
the  two  frequencies.  As  this  wave-form  reaches  the  ear,  the 
membrane  closing  it  does  not  vibrate  regularly  ;  at  certain 
intervals  it  is  violently  stimulated,  while  at  times  in  between 
it  is  not  moved  at  all.  This  necessarily  produces  in  the  ear 
a  sensation  analogous  to  that  caused  in  the  eye  by  an  inter- 
mittent series  of  flashes  of  light ;  and  both  these  sensations 
are  disagreeable.  The  degree  of  the  unpleasantness  de- 
pends upon  the  number  of  stimuli  given  the  ear  or  eye  in 
a  given  time,  and  also  upon  the  frequencies  of  the  waves. 
In  sound  these  unpleasant  effects  are  called  "  beats  ; "  and, 
if  they  are  produced  by  the  interference  of  two  waves 
whose  frequencies  are  n  and  nr,  there  will  be  n'  —  n  beats 
per  second,  and  the  unpleasantness  will  vary  with  n'  —  n 
and  also  n.  That  is,  a  certain  number  of  beats  per  second 
may  be  more  disagreeable  at  one  pitch  than  another.  If 
the  number  of  beats  exceeds  forty  or  fifty  in  a  second,  it 
becomes  difficult  to  detect  them ;  and  so  the  most  disagree- 
able effect  is  produced  when  the  pitches  of  the  two  sounds 
are  nearly  the  same.  If  the  pitches  are  identically  the 


133]  SOUND   SENSATION  165 

same,  there  are,  of  course,  no  beats  ;  and  their  absence  may 
serve  to  demonstrate  the  unison  of  two  notes. 

133,  Differential  and  Summational  Notes.  When  two  vi- 
brating bodies  of  frequencies  n  and  n1  are  set  in  motion  vio- 
lently, other  sounds  than  those  of  pitches  n  and  n'  are  heard, 
if  the  same  volume  of  air  is  agitated  by  the  two  vibrations. 
Among  these  so-called  "  combination-notes  "  are  those  of 
pitch  n  -f  n'  and  n  —  n1,  which  are  called  respectively  "  sum- 
mational  "  and  "  differential "  notes.  Other  notes  are  also 
heard  under  different  conditions. 


CHAPTEE  IV 

EEFLECTION  AND  EEFRACTION 

134.  Introduction.  Waves  of  all  kinds,  moving  in  any 
medium,  suffer  certain  changes  when  they  reach  the  bound- 
ing surface  which  separates  that  medium  from  another. 
When  the  waves  reach  the  boundary,  they  produce  "  re- 
flected "  waves  back  into  the  first  medium,  and  also  cause 
waves  to  pass  into  the  second  medium.  Those  waves  in 
which  these  facts  are  most  easily  observed  are  light-weaves, 
where  the  paths  of  the  waves  may  be  seen ;  and  their  prop- 
erties will  be  studied  later.  But  every  one  is  acquainted 
with  a  few  of  the  fundamental  laws  of  light :  the  angle 
of  incidence  equals  the  angle  of  reflection  ;  when  light- 
waves pass  from  one  medium  into  another,  the  direction  of 
the  waves  is  changed,  i.  e.  suffers  "  refraction ; "  concave 
mirrors  and  convex  lenses  can  focus  the  light-waves  ;  there 
are  certain  so-called  interference  and  diffraction  phenomena. 
All  these  properties  are  common  also  to  those  waves  in 
ordinary  matter  which  can  produce  sound,  i.  e.  to  all  com- 
pressional  waves. 

That  sound  waves  can  be  reflected  is  made  evident  by 
the  existence  of  echoes  and  "  whispering-galleries."  Con- 
cave mirrors  can  also  easily  focus  sound  waves ;  and  a 
"sounding-board"  is  an  illustration  of  this. 

Lenses  can  also  be  made  of  suitable  size,  so  that  the 
waves  from  a  vibrating  body  placed  at  A  will  all  converge, 
after  passing  through  the  lens,  at  some  point,  B.  A  and  B 
are  known  as  "  conjugate  foci."  (See  Fig.  115). 


135] 


KEFLECTION  AND  REFRACTION 


167 


FIG.  115. 

135.  Reflection.  The  nature  of  the  reflection  of  a  train 
of  waves  of  any  kind  depends  on  the  differences  between 
the  two  mediums  at  whose  boundary  the  reflection  takes 
place.  This  can  best  be  explained  by  some  mechanical 
illustrations.  Consider  two  series  of  spheres  arranged  in 


FIG.  116. 


line  and  separated  by  spiral  springs.  Let  the  masses  of 
the  spheres  and  the  elasticity  of  the  springs  be  such  that 
the  velocity  of  a  compressional  wave  is  different  in  the 


168  THEORY  OF  PHYSICS  [CH.  IV 

two  series.  Call  one  the  "  fast "  series ;  the  other,  the 
"  slow." 

Now,  if  the  extreme  sphere  of  the  slow  series  is  pushed 
in  so  as  to  compress  the  spring  attached  to  it,  a  compression 
is  sent  along  the  row ;  but,  when  the  wave  reaches  the  end 
of  this  series,  the  velocity  of  the  compression  in  the  fast 
series  is  such  that  the  last  sphere  of  the  slow  series  does  not 
meet  with  enough  opposition  ;  and  so  it  swings  further  than 
it  should,  and  thus  stretches  the  spring  between  it  and  its 
next  neighbor  in  the  slow  series.  Consequently  a  rarefac- 
tion is  sent  back  as  a  reflected  wave  along  the  slow  series 
of  spheres.  But,  if  a  compression  is  sent  along  the  fast 
series,  when  the  last  sphere  is  reached,  it  compresses  the 
spring  between  it  and  the  first  sphere  of  the  slow  series ; 
and,  since  the  velocity  in  this  series  is  less  than  along  the 
other,  the  spring  does  not  move  fast  enough,  and  so  be- 
comes unduly  compressed.  Consequently  by  its  reaction 
on  the  fast  series  a  compression  is  sent  back  as  the  re- 
flected wave.  Similarly,  if  a  rarefaction  is  sent  along  the 
slow  series,  a  condensation  is  reflected ;  and  if  it  is  sent 
along  the  fast  series,  a  rarefaction  is  reflected.  So.  if  a 
succession  of  condensations  and  rarefactions  is  sent  along 
either  series,  a  train  of  waves  is  reflected. 

A  special  case  of  a  series  of  spheres,  whose  velocity  of 
transmission  is  slow,  is  one  where  there  is  no  transmis- 
sion at  all,  e.  g.  the  last  sphere  of  one  series  is  fastened  to 
a  rigid  wall.  In  this  case  the  reflected  wave  is  identical 
with  the  incident,  only  going  in  the  opposite  direction. 
A  special  case  of  a  series  of  spheres  whose  velocity  of 
transmission  is  very  rapid,  i.  e.  where  little  opposition  is 
offered  to  the  incident  wave,  is  when  there  is  no  second 
series  at  all ;  that  is,  there  is  a  single  series  of  spheres,  and 
the  last  sphere  is  not  connected  with  anything  except  the 
spring  which  gives  it  motion.  In  this  case  the  reflected 
train  of  waves  differs  from  the  incident  in  that  a  condensa- 
tion is  replaced  by  a  rarefaction,  and  vice  versa. 


137]  REFLECTION  AND  REFRACTION  169 

Transverse  Waves  in  Cords.  Another  illustration  of  the 
reflection  of  waves  is  given  when  transverse  waves  are 
sent  along  a  rope  or  stretched  string.  Consider  the  two 
cases,  (1)  reflection  from  a  fixed  point,  i.  e.  the  further  end 
is  rigidly  fastened ;  (2)  reflection  from  a  "  free  "  end,  i.  e. 
the  further  end  is  not  fastened  to  anything. 

136.  1.  Reflection  from  a  Fixed  Point.  Let  a  transverse 
harmonic  wave  be  sent  along  a  cord  whose  further  end  is 
fastened  to  a  wall  or  any  fixed  support.  At  the  instant  it 
reaches  the  fixed  end,  it  may  be  represented  as  in  the 
figure,  where  the  dotted  line  shows  the  original  position  of 
the  cord.  B  is  the  fixed  point,  and  A  is  the  end  which  is 
moved  transversely  with  a  harmonic  vibration.  The  in- 
stant this  wave  reaches  B,  a  similar  wave  starts  back  in 


FIG.  117. 

the  opposite  direction  along  the  cord,  so  that,  at  later  in- 
stants, as  the  end  A  is  vibrated,  thus  sending  out  waves  in 
the  cord,  there  are  two  trains  of  waves  moving  in  opposite 
directions,  —  one  direct,  the  other  reflected.  These  two 
trains  have  the  same  frequency  and  velocity  ;  and  their  posi- 
tions are  shown  in  the  following  diagrams,  at  instants  of 
time  differing  by  one  eighth  of  the  period  of  the  wave.  The 
faint  continuous  line  represents  the  direct  wave  ;  the  dot- 
ted line,  the  reflected  wave ;  the  heavy  line,  the  resulting 
motion  produced  by  the  sum  of  the  displacements  due  to 
each  of  the  two  waves  separately. 

137.  It  is  to  be  noticed  that  at  regular  intervals  there 
are  points  of  the  cord  which  never  move,  the  two  waves 
neutralizing  each  other.  These  points  are  called  "  nodes  ;  " 
and  the  distance  between  consecutive  nodes  is  seen  at  once 
to  be  one  half  a  wave-length  of  either  train  of  waves. 
Also,  in  between  the  nodes,  it  is  noticed  how  the  portions 
of  the  cord  vibrate  as  a  whole  across  the  original  direction, 

o*  O 


170 


THEORY   OF  PHYSICS 


[CH.  IV 


The  middle  point  of  each  such  vibrating  portion  is  called 
a    "  loop,"   and    the    distance    between    two    consecutive 

loops  is  also  one  half 
a  wave-length.    There 
will  thus  be  produced 
in  the  cord,  as  a  con- 
sequence of  the  super- 
position of  these  two 
waves,     a    transverse" 
vibrAtion.      This     is 
called  a  "stationary" 
vibration,  because,  al- 
though at  any  instant 
the  cord  has  the  ap- 
pearance of  a  train  of 
waves,   this    apparent 
wave  does  not  advance ; 
the   crests  remain   at 
the   same   points,   die 
down,  and   then  hol- 
lows take  their  places, 
while  the  hollows  rise 
and     become     crests ; 
-j£*'-Ii^-^L^^r^-       and  so  on. 

^.  ^--T        T^=~^  -T=^S\ 

If  a  cord  is  thus  set 
in  stationary  vibration 
by  the  hand,  it  is  no- 
ticed at  once  that  the 
end  of  the  cord  in  the 
hand  is  approximately 
a  node.  It  does  move 
some,  because  the  en- 
ergy from  the  hand  is 
here  given  the  cord;  but 
its  motion  is  not  nearly 
FlG  118  so  great  as  at  a  loop. 


138]  REFLECTION  AND  REFRACTION  171 

138.  Thus,  in  a  cord  fastened  at  one  end,  and  set  in 
stationary  vibrations  by  a  hand  at  the  other,  there  are 
nodes  at  each  end.  (So  it  is  also  if  a  cord  is  fastened  at 
both  ends  and  set  in  vibration  by  some  blow.)  Conse- 
quently, there  must  be  a  whole  number  of  vibrating  por- 
tions or  segments  in  the  cord.  If  there  are  .2V  -f  1  nodes, 
counting  the  two  ends,  there  are  N  such  vibrating  seg- 
ments. Let  v  be  the  velocity  of  the  two  trains  of  waves 
in  the  cord  ;  n,  the  frequency  ;  X,  the  wave-length  ;  Z,  the 
length  of  the  string.  Then 

v  —  n\;   L  —  N  -, 

since  the  distance  between  two  nodes  is  -  .    Consequently. 

Nv 


Therefore,  since  N  can  equal  1,  2,  3,  4,  etc.,  there  are  cer- 
tain definite  values  of  n  which  can  produce  stationary 
vibrations  in  a  given  cord.  To  make  N  greater,  —  that 
is,  to  increase  the  number  of  vibrating  segments,  —  it  is 
necessary  to  increase  n,  the  number  of  vibrations  per 
second  given  the  cord. 

If  the  tension  in  the  cord  is  increased  by  some  longi- 
tudinal force,  the  velocity,  v,  will  be  increased.  Conse- 
quently, if  n  is  not  changed,  and  if  the  tension  in  the 
cord  is  increased  enough,  N  must  decrease  by  1.  That  is, 
if  a  cord  is  vibrating  with  4  segments,  it  will  change  into 
3  segments,  if  the  tension  is  increased  sufficiently. 

All  these  deductions  can  be  easily  verified  by  direct  ex- 
periments. 

139.  2.  Reflection  from  a  Free  End.  Let  a  transverse 
harmonic  train  of  waves  be  sent  along  a  cord  whose  far- 
ther end  is  free.  The  reflected  waves  will  have  the  same 
frequency  and  velocity,  and  they  will  be  identical  with  the 
reflected  waves  from  a  fixed  point,  except  that  a  crest  takes 


172 


THEORY   OF  PHYSICS 


[CH.  IV 


the  place  of  a  hollow  and  vice  versa.     The  following  draw- 
ings represent  the  direct  train  of  waves,  the  reflected  train, 

and  the  resulting  motion, 
at  intervals  of  time  equal 
to  one  eighth  of  a  period. 

It  is  seen  that  here  also 
there  are  nodes  and  loops, 
i.  e.  a  stationary  vibration  ; 
and  the  distance  between 
two  nodes  or  between  two 
loops  is  equal  to  one  half 
the  wave-length  of  either 
the  direct  or  the  reflected 
waves.  There  is  a  loop,  of 
course,  at  the  free  end  of 
the  cord. 

140.  Longitudinal  Waves. 
Just  as  in  the  ball  and 
spring  model  discussed  in 
Article  135,  so  longitudinal 
waves  may  be  sent  along  a 
stretched  cord,  a  solid  rod,  a 
column  of  air.  These  waves 
are,  of  course,  compressional 
ones,  and  so  move  with  the 
"velocity  of  sound"  in  those 
substances.  The  waves  will 
be  reflected  from  the  ends  of 
the  medium ;  and  thus  it 
will  be  set  in  stationary  vi- 
bration. If  the  end  is  fixed, 
there  will  be  a  node  there ; 
if  it  is  free,  there  will  be  a  loop.  The  number  of  vibrating 
segments  depends  upon  the  velocity  of  the  waves  and  the 
number  of  vibrations  per  second. 


L i_J^rtrl_^_ 


CHAPTEE  V 

VIBRATING   BODIES 

141.  NEARLY  every  musical  instrument  when  it  is  pro- 
ducing the  compressional  waves  commonly  called  sound- 
waves is  making  stationary  vibrations,  the  nature  of  which 
has  been  discussed  in  the  previous  chapter.     The  funda- 
mental property  of  stationary  vibrations  is  the  existence 
of  nodes  and  loops  which  may  be  regarded  as  produced  by 
the  superposition  of  two  identical  trains  of  waves  in  oppo- 
site directions ;   and  it  is  a  general  law  that  the  wave- 
length of  both  the  direct  and  reflected  waves  is  twice  the 
distance  between  two  nodes   or  between  two  loops.     As 
explained  also,  any  body  such  as  a  cord  or  a  column  of 
air  can  vibrate  in  many  ways,  with  one  segment  or  two  or 
three,  etc. ;  and  so  in  general,  unless  special  precautions 
are  taken,  a  body  set  in  vibration  at  random  will  have  a 
motion  which  is  compounded  of  several  of  these  different 
simple  modes  of  vibration.      Some  special  cases  will  now 
be  considered. 

142.  Transverse  Vibrations  of  a  Uniform  Stretched  Cord. 
When  a  cord  fastened  at  both  ends  vibrates  transversely, 
it  itself  does  not  affect  greatly  the  surrounding  air,  owing 
to  its  small  cross-section.     But  the  supports  to  which  the 
ends  are  fastened  are  never  quite  rigid ;  and  so  they  also 
vibrate,  and  in  turn  produce  vibrations  in  the  board  to 
which  they  are  joined.     This  board  may  be  made  broad ; 
and  consequently  its  vibrations  will   produce  a   marked 
effect  on  the  surrounding  air.     This  explains  the  action 
of   a  violin  or  a  piano,   where   the  wooden   portions   are 
much  more  important  than  the  strings. 


174 


THEORY  OF  PHYSICS 


[CH.  V 


There  are  many  ways  in  which  a  string  thus  stretched 
can  be  set  in  transverse  vibrations,  —  it  may  be  plucked 
one  side,  it  may  be  rubbed  with  a  violin  bow,  or  it  may  be 
struck  a  blow.  There  are  also  many  possible  modes  of 
vibration.  The  simplest  one  is  when  the  only  nodes  are 
at  the  two  ends.  In  this  case,  if  L  is  the  length  of  the 
string,  n  the  frequency,  v  the  velocity  of  transverse  waves 
in  the  string,  the  wave-length,  v/n,  equals  2  L.  That  is, 

n  =  v  1 2  Z. 

This  vibration   is   called  the   "fundamental."     The   next 
simplest  mode  of  vibration  is  when  the  string  is  divided 


FIG.  120. 


into  two  vibrating  segments,  there  being  a  node  at  the 
middle  point.     If  ni  is  the  frequency  of  the  vibration, 

v  I  ni  =  L, 
or  HI  =  v  I  L. 

This  vibration  is  called  the  "  first  partial ; "  and  its  fre- 
quency is  just  twice  that  of  the  fundamental.     The  next 


FIG.  121. 


simplest  case  is  when  there  are  three  vibrating  segments. 
If  nz  is  the  frequency,  it  is  obvious  that 

v/nz  =  2L/3, 
or  nz  =  3  v  /  2  L. 


142]  VIBRATING  BODIES  175 

This  vibration  is  called  the  "  second  partial  ;  "  and  its  fre- 
quency is  three  times  that  of  the  fundamental.  In  a 
similar  manner,  it  is  easily  seen  that  there  are  other  possi- 
ble partials,  whose  frequencies  are  four,  five,  etc.  times  that 
of  the  fundamental. 

Corresponding  to  each  of  these  partial  vibrations  there  is 
a  definite  sound-sensation,  whose  pitch  is  given  by  the  fre- 
quency of  the  vibration  ;  and,  if  the  string  is  set  in  motion 
at  random,  several  of  these  sounds  may  be  heard.  Of 
course  the  point  of  the  string,  at  which  is  struck  the  blow 
producing  the  vibration,  is  always  a  loop  in  the  resulting 
vibration  ;  and  so  any  partial  vibration  which  would  make 
it  necessary  for  that  point  to  be  a  node  cannot  occur,  and 
the  corresponding  sound  is  absent.  Thus,  if  a  string  is 
set  in  vibration  by  a  blow  at  its  middle  point,  the  first, 
third,  fifth,  etc.  partials  cannot  arise.  Since  the  quality 
of  any  sound  depends  upon  the  number  and  intensity  of 
the  partials  present  with  the  fundamental,  the  sound  pro- 
duced by  the  transverse  vibrations  of  a  stretched  string 
depends  largely  upon  the  point  where  it  is  struck  or 
bowed. 

It  is  proved  by  theory  and  verified  by  experiment  that 
the  velocity  of  a  transverse  wave  in  a  thin,  uniform  stretched 
string  is 


<rp 


where  P  is  the  tension  in  the  string,  that  is,  the  stretching 
force  ;  cr  is  the  area  of  the  cross-section  of  the  string  ;  p  is 
the  density  of  the  string.  Consequently  the  pitch  of  the 
fundamental,  n  =  v  /  2  Z,  is 


and  so,  as  the  tension  is  increased,  the  pitch  rises. 


176  THEORY  OF  PHYSICS  [CH.  V 

143.  Longitudinal  Vibrations  in  a  Stretched  Cord  or  Wire. 
This  mode  of  vibration  may  be  produced  by  rubbing  the  cord 
in  the  direction  of  its  length  with  a  damp  cloth.  The 
waves  thus  excited,  which  by  their  combination  produce  the 
vibration,  are  compressional  ;  and  so  the  velocity  of  the 
waves  is  what  is  called  the  velocity  of  sound  in  the  material 
of  which  the  cord  or  wire  is  made. 

The  simplest  case,  as  before,  is  that  where  the  only  nodes 
are  at  the  two  ends.  If  L  is  the  length  of  the  cord  or  wire, 
n  the  frequency,  V  the  velocity  of  sound  in  the  given 
material, 

r/n  =  2  L, 

and  so  n  =    V  /  2  L. 

Since  n  and  L  can  both  be  measured,  this  gives  a  method 
of  determining  V,  the  velocity  of  sound  in  the  given  sub- 
stance. Other  cases  of  vibration,  where  there  are  nodes 
along  the  cord  or  wire,  are  obtained  with  difficulty. 

It  may  be  proved  that,  in  this  case, 


where  E  is  Young's  modulus  of  elasticity  (see  Art.  83), 
and  p  is  the  density  of  the  cord  or  wire.  Then  the  fre- 
quency of  the  fundamental  vibration  is 


or 

And  since  Z,  n,  and  p  are  all  easily  measiired  for  any  given 
cord  or  wire,  this  gives  a  simple  method  for  the  determina- 
tion of  Young's  modulus.  It  should  be  noticed  that  the 
frequency  of  the  vibration,  for  a  definite  length,  does  not 
depend  upon  the  cross-section  of  the  cord  or  wire,  nor 
upon  the  tension  in  it,  provided  that  this  is  not  great 
enough  to  influence  Young's  modulus  or  the  density. 


145] 


VIBRATING  BODIES 


177 


144.  Longitudinal  Vibrations  of  Rods.  This  mode  of  vi- 
bration may  be  produced  by  rubbing  the  rod ;  and  it  is 
exactly  similar  to  the  preceding  case,  with  this  exception : 
a  stretched  cord  or  wire  is  fastened  at  its  two  ends,  while  a 
rod  is  not  in  general. 

The  simplest  longitudinal  vibration  of  a  rod  is  when  it 
is  clamped  at  its  middle  point.  There  is  then  a  loop  at 
each  end  ;  and  the  wave-length 

r/7*  =  2z, 

or  n=    V/2L. 

Here,  also,  n  and  L  may  be  measured ;  and  so  this  gives  a 
method  for  the  measurement  of  the  velocity  of  sound  in 
the  material  of  which  the  rod  is  made. 


FIG.  123. 


FIG.  122. 

145.    Transverse  Vibrations  of  a  Rod.      The 

rod  may  be  clamped  at  some  point,  and  then 
struck  a  blow  at  one  end  perpendicular  to  its 
length.  One  of  the  commonest  modes  of  such 
a  vibration  is  when  one  end  is  clamped,  and 
the  other  vibrates  freely.  In  this  case  there 
is  a  loop  at  the  latter  end  and  a  node  at  the 
former.  Another  mode  of  vibration  of  a  rod 
is  when  there  are  two  nodes  at  equal  distances 
from  the  ends.  An  illustration  of  this  is  an 
ordinary  tuning-fork,  which  is  simply  a  rod  so 
bent  as  to  form  a  U,  and  which  has  an  arm 
attached  at  the  bottom  of  its  curved  portion. 
When  a  tuning-fork  vibrates,  its  two  ends  are 
loops ;  and  the  attached  arm  is  also  in  violent  motion,  as 
may  be  proved  by  resting  it  on  a  table  or  box  ;  for  the  latter 
is  immediately  set  in  vibration. 


178 


THEORY  OF  PHYSICS 


[CH.  V 


146.  Vibrations  of  Columns  of  a  Gas.     The  only  vibrations 
possible  in  a  column  of  gas  are,  of  course,  longitudinal  ones, 
becaiise  a  gas  has  no  rigidity.     These  vibrations 

are  extremely  common  in  musical  instruments, 
e.  g.  organ-pipes,  horns,  flutes,  etc. ;  and  they 
are  easily  produced.  One  way  is  simply  to 
blow  over  the  opening  of  a  hollow  tube;  another 
is  to  hold  some  vibrating  instrument,  like  a 
tuning-fork,  over  the  opening.  In  an  ordinary 
organ-pipe  air  is  driven  in  through  a  narrow 
passage,  and,  striking  the  lip  at  the  bottom  of 
the  pipe,  produces  a  periodic  motion  of  the  air 
near  there,  which  sets  in  vibration  the  column 
of  air  in  the  pipe.  There  is  always  a  loop  at 
the  lip  of  the  organ-pipe,  owing  to  the  violent 
motion  near  it.  Organ-pipes  are  of  two  kinds, 
—  closed  and  open. 

147.  1.  Closed  Pipes.     The   end  of  the  pipe 
being  closed,  there  is,  of  course,  a  node  at  that 
point. 

The  simplest  mode,  then,  of  vibration  is  whan  there  is  a 
node  at  the  closed  end  and  a  loop  at  the  lip,  there  being 


FIG.  124. 


FIG.  125. 


neither  node  nor  loop  in  between.     The  wave-length  is 
four  times  the  distance  from  a  node  to  a  loop.     So,  if  L 
•  is  the  length  of  the  pipe,   V  the  velocity  of  sound  waves 
in  the  gas  enclosed,  and  n  the  frequency, 


or  n  —  v  '  4 

This  is  the  fundamental  vibration. 


148] 


VIBRATING  BODIES 


179 


The  next  simplest  mode  of  vibration  is  when  there  are  a 
node  and  a  loop  in  between  the  two  ends.  (It  is,  of  course, 
impossible  to  have  two  nodes  without  there  being  a  loop 


! 

*-                   ! 

L                                  N 
FIG.  126. 

L 

N 

between  them.)     In  this  case  the  distance  between  two 
nodes  is  2  L  /  3.     Hence,  if  nz  is  the  frequency, 


V/n*  =  4Z/3, 


or 


This   is   evidently   the   second   partial,  the   first  one  not 
occurring. 


^^-  —               ^-^_ 

'  .  " 

i 

""*--^^_                  ^^^^^ 

1 

*  

L                r 

1                      L                      \ 
FIG.  127. 

1                      L                      N 

The  next  simplest  mode  of  vibration  is  when  there  are 
two  nodes  and  two  loops  in  between  the  ends.  If  nt  is  the 
frequency, 


or 


This  is  the  fourth  partial,  the  third  not  occurring. 

So  the  other  modes  of  vibration  may  be  discussed.  It 
is  evident  that  in  a  closed  organ-pipe  only  the  fundamental 
and  the  alternate  partial  vibrations  can  occur. 

148,  2.  Open  Pipes.  In  an  open  pipe  there  is  a  loop  at 
each  end.  (In  reality,  owing  to  the  inertia  of  the  air,  the 
loops  do  not  come  exactly  at  the  ends,  but  at  a  slight  dis- 


180 


THEORY  OF  PHYSICS 


[CH.  V 


tance  beyond  them,  depending  largely  upon  the  radius  of 
the  pipe  and  the  length  of  the  waves. ) 


N 
FIG.  128. 


The  simplest  mode  of  vibration  of  an  open  pipe  is  when 
there  is  a  loop  at  each  end  and  one  node  in  between.  If 
n  is  the  frequency, 


or 


That  is,  the  lowest  frequency  of  an  open  pipe  is  twice  that 
of  a  closed  pipe. 


I  -^ 

—                                        — 

I 

L                           |> 

J                             L                            !> 
FIG.  129 

J                             L 

The  next  simplest  mode  of  vibration  is  when  there  are 
two  nodes  and  one  loop  in  between  the  ends.  If  Wi  is  the 
frequency, 


or 


1               "*" 

1 

<                    (                   > 

^~~j-^ 

1 
i 

L                  N                   L 

N                   L                  N                  L 
FIG.  130. 

The  next  simplest  mode  of  vibration  gives 

7^  =  2  T/Z, 
and  so  on. 


150]  VIBRATING   BODIES  181 

It  is  evident  that  in  open  organ-pipes  the  fundamental 
and  all  the  partials  may  occur. 

The  quality  of  the  sound  heard  as  the  result  of  the 
vibrations  of  the  columns  of  gas  in  any  organ-pipe  depends 
upon  the  nature  of  the  partials  present.  When  an  open- 
ing is  made  in  the  side  of  a  pipe,  as  in  a  horn  or  a  flute, 
a  loop  must  occur  at  that  point ;  and  so  the  pitch  and 
quality  of  the  resulting  sound  are  changed.  By  changing 
the  motion  of  one's  lips  at  the  mouthpiece  of  a  horn,  differ- 
ent vibrations  may  be  produced. 

Further,  since  in  a  closed  organ-pipe  the  fundamental 
vibration  has  a  frequency  n  =  V/  4  L,  and  since  n  and  L 
can  both  be  measured,  V,  the  velocity  of  sound  in  the  gas 
filling  the  pipe,  may  be  easily  determined.  It  is  found,  as 
might  be  expected,  that  the  velocity  in  the  pipe  depends 
somewhat  upon  the  size  of  the  pipe  and  also  upon  the 
wave-length. 

149.  Vibrations  of  Plates  and  Membranes.     A  plate  or  a 
stretched  membrane  may  easily  be  set  in  transverse  vibra- 
tions by  rubbing  a  violin  bow  across  its  edge.     In  general 
there  will  be  certain  lines  in  the  plate  or  membrane,  where 
there  is  no  motion.     These  lines  are  called  nodal  lines ; 
and  their  position  may  be  found  by  having  light  sand  or 
powder  sprinkled  over  the  surface.     The  sand  will  move 
away  from  the  places  of  greatest  motion  and  be  heaped  up 
along  the  nodal  lines.     As  might  be  expected,  the  position 
and  form  of  the  nodal  lines  depend  upon  where  the  plate 
or  membrane  is  held,  and  where  and  how  it  is  rubbed. 

150.  Vibrations  of  Bells.    A  bell-shaped  vessel,  like  a  bowl 
of  any  kind,  may  have  two  kinds  of  vibrations,  transverse 
and  longitudinal.     The  transverse  ones  may  be  produced 
by  striking  the  bowl  a  blow  or  by  rubbing  across  its  edge. 
The  longitudinal  ones  may  be  produced  by  rubbing  along  its 
edge.     As  a  rule,  most  actual  bells,  especially  large  ones, 
are  not   uniform,  but  have   what  may  be  considered   as 
thicker  places  at  certain  points.     If  this  thick  place  hap- 


182  THEORY  OF  PHYSICS  [CH.  V 

pens  to  be  at  a  node,  there  will  be  a  certain  definite  fre- 
quency; while,  if  it  comes  at  a  loop,  the  frequency  will 
be  slightly  different.  So,  if  the  bell  is  struck  at  random, 
both  these  vibrations  will  occur,  and  beats  (see  Art.  132) 
will  be  heard. 

151.  The  Human  Voice.     The  vibrations  of  the  lips,  the 
tongue,  and  the  so-called  "  vocal  cords,"  or  "  larynx,"  pro- 
duce the  human  voice.     The  pitch  depends  upon  the  fre- 
quency of  the  vibration.     Various  partials  always  occur ; 
and  these    may  be   strengthened  or  modified   by  holding 
the  lips  and  tongue  in  certain    positions.      The  so-called 
"vowel"  sounds  are  due  to  the  simple  vibration    of    the 
larynx  and  the  modification  of  the  partials  by  the  shape 
given  the  mouth.      The  "  consonant "  sounds  are  due  to 
the  vibrations  of  the  lips  and  tongue  in  different  positions. 

152.  It  should  be  noted  that  in  all  the  cases  and  methods 
of  vibration,  which  have  been  discussed,  the  wave-length 
of  the  sound  waves  varies  directly  as  the  length  of  the 
vibrating  segment  which  produces  the  waves.     This  is  a 
general  property  of  waves :  the  wave-length  varies  directly 
as  the  linear  dimension  of  the  vibrating  body. 


CHAPTER   VI 

VELOCITY  OF   SOUND 

WHAT  is  meant  by  the  "  velocity  of  sound  "  is  the  ve- 
locity of  compressional  waves  in  a  material  medium  of  a 
given  kind  and  under  given  conditions. 

153.  Direct  Method.     This  velocity  may  be  determined 
without  much  difficulty  by  a  direct  measurement  if   the 
expanse  of  the  medium   is  great  enough.     Thus  a  pistol 
may  be  fired  in  the  air,  or  a  bell  rung  under  water  ;  and 
the  interval  of  time  which  elapses  between  this  instant 
and  that  when  the  corresponding  sound  is  heard  at  a  meas- 
ured distance  away  may  be  noted.     The  ratio  of  this  dis- 
tance to  the  interval  of  time  gives  the  velocity  of  sound  in 
air  or  water  under  the  existing  conditions. 

154.  Indirect  Method.     Another  method  is  to  apply  the 
formula  stated  in  Article  119, 


where  E  is  the  bulk  modulus  and  p  is  the  density,  Thus, 
if  E  and  p  are  known  (and  they  can  in  general  be  deter- 
mined), V  can  be  at  once  calculated.  For  a  gas,  as  proved 
in  Article  119, 

V  '  = 

and,  if  V  can  be  measured  by  any  means  for  the  gas,  this 
formula  may  serve  for  the  calculation  of  any  of  the  other 
quantities.  It  should  be  noted  that  T  =  273  +  t°  centi- 
grade ;  and  so,  as  the  temperature  of  a  gas  increases,  the 
velocity  of  sound  in  it  does  also. 


184  THEORY  OF  PHYSICS  [CH.  VI 

155.  Velocity  in  a  Column  of  Gas,  In  the  last  chapter 
other  methods  for  the  measurement  of  the  velocity  of  sound 
in  any  medium  have  been  explained.  In  particular,  to 
measure  the  velocity  of  sound  in  any  gas,  enclose  a  column 
of  the  gas  in  a  pipe  open  at  one  end,  vibrate  a  tuning-fork 
of  known  frequency  over  this  end,  and  alter  the  length  of 
the  pipe  by  means  of  a  movable  piston  until  the  column  of 
gas  is  vibrating  in  unison  with  the  fork.  Let  n  be  the  fre- 
quency of  the  vibration,  L  the  length  of  the  column  of 
gas,  and  V  the  velocity  of  sound  in  the  gas ;  then,  if  the 
vibration  of  the  column  of  gas  is  its  fundamental  one, 

4Z=  V/n,    •  ' 
or  F=4rcZ. 

An  upper  partial  vibration  of  the  column  of  gas  might  have 
been  used ;  and,  by  suitable  methods,  all  error  due  to  the 
displacement  of  the  loop  from  the  open  end  of  the  pipe 
may  be  obviated. 

If  the  velocity  of  sound  in  any  one  gas  is  known,  it  may 
be  determined  for  any  other  gas  in  a  most  simple  manner. 
Fill  two  organ-pipes  with  the  two  gases,  and  set  the  col- 
umns in  vibration.  Then  the  length  of  one  can  be  altered 
until  the  pitches  of  the  two  sounds  are  identical.  If  the 
pipes  are  closed  at  one  end  and  are  emitting  their  funda- 
mentals, and  if  n  is  the  frequency  common  to  both  pipes,  Vl} 
and  Vz  the  two  velocities,  LI  and  Z2  the  lengths  of  the  two 
columns, 

Fi         F2 

"TA'Iz;' 

and  hence  Fi  :  F2  =  Zi  :  Z2. 

So  the  unknown  velocity  may  be  at  once  calculated. 

As  noted  before,  it  will  be  found  that  the  velocity 
changes  with  the  temperature,  and  that  it  depends  slightly 
upon  the  relative  dimensions  of  the  wave-length  and  the 
organ-pipe. 


158]  VELOCITY  OF  SOUND  185 

156.  Velocity  in  a  Column  of  a  Liquid.     Stationary  waves 
may  also  be  produced  in  a  closed  pipe  filled  with  a  liquid, 
by  causing  the  piston  closing  one  end  to  vibrate  harmoni- 
cally.    If  n  is  the  frequency,  L  the  distance  between  two 
nodes,  and  V  the  velocity  of  sound  in  the  liquid, 

2L=V/  n, 
or  V=2nL. 

It  is  not  difficult  to  determine  accurately  the  position  of 
the  nodes;  because,  if  any  light  powder  is  suspended  in 
the  liquid,  it  will  arrange  itself  in  the  tube  so  that  the 
nodes  and  loops  may  be  easily  distinguished. 

157.  Velocity  in  a  Solid  Wire  or  Rod.     The  velocity  of 
sound  in  a  solid  wire  or  rod  may  be  determined  by  simply 
setting  it  in  longitudinal  vibration.      If  the  wire  has  a 
length  L,  and  if  it  is  stretched  between  two  fixed  clamps, 
then 

2L  =  V/n, 

and  V  =  2  n  L. 

The  velocity  of  sound  in  two  different  wires  may  be  also 
thus  compared,  by  altering  the  length  of  one  until  its  fre- 
quency equals  that  of  the  o'ther.  If  V\  and  F2  are  the  two 
velocities,  L\  and  Z2  the  two  lengths, 

Fx  :  Fa  =  A  :  Za. 

If  a  rod  is  clamped  at  its  middle  point,  and  set  in  longi- 
tudinal vibrations,  there  are  loops  at  the  two  ends  ;  so,  if 
L  is  the  length  of  the  rod, 


or  = 

158.  Kundt's  Method.  Another  indirect  method  of  de- 
termining the  velocity  of  sound  in  any  medium  —  solid, 
liquid,  or  gas  —  is  known  as  Kundt's  method,  from  the  name 


186  THEORY  OF  PHYSICS  [CH.  VI 

of  the  physicist  who  devised  it.  Its  principle  is  to  compare 
the  unknown  velocity  with  one  that  is  known.  The  ap- 
paratus consists  of  two  parts :  a  glass  tube,  about  100  cm. 
long  and  of  2  or  3  cm.  diameter,  which  is  closed  at  one 
end  by  a  tight-fitting  piston,  A,  and  at  the  other  by  a 
loosely  fitting  piston,  B ;  and  a  solid  rod,  perhaps  100  cm. 
long,  which  is  firmly  clamped  at  its  middle  point,  coaxial 
with  the  glass  tube,  and  which  carries  on  one  end  the 
piston  B.  The  solid  rod  is  set  into  longitudinal  vibrations 


A  B  W 

FIG.  131. 

by  being  rubbed  lengthwise  with  a  dampened  cloth ;  and 
the  vibrations  of  the  piston  B  set  into  vibration  the  gas 
enclosed  in  the  glass  tube.  These  vibrations  in  the  gas 
will  not  become  stationary  unless  the  length  of  the  column 
of  gas  is  such  as  to  equal  a  whole  number  of  vibrating 
segments.  Some  light  powder  is  placed  in  the  tube,  and,  if 
the  vibrations  are  not  stationary,  the  powder  will  not  take 
any  permanent  distribution.  But,  by  moving  the  piston 
A,  the  length  of  the  column  of  gas  may  be  so  changed  that 
the  vibrations  will  become  stationary.  This  will  be  evi- 
dent by  the  arrangement  of  the  powder  in  the  tube,  and 
the  distance  between  two  nodes  may  be  accurately  meas- 
ured. (It  is  to  be  noted  that  the  point  B  is  a  node  for  the 
gas,  although  it  is  a  loop  for  the  rod ;  just  as,  when  a  rope  is 
set  in  vibration  by  the  motion  of  one's  hand,  the  end  in 
the  hand  is  a  node.)  If  LI  is  the  length  of  the  rod,  FI  the 
velocity  of  sound  in  the  rod,  Lz  the  average  distance  be- 
tween two  nodes  in  the  column  of  gas,  F2  the  velocity  of 
sound  in  the  gas,  n  the  frequency  of  the  vibration,  which  is 
the  same  for  both  rod  and  gas, 

2Z1=  Fi/n;   2  Z2  -  F2  / n, 
or  Fx  :  F2  =  Zi  :  Z2. 


158] 


VIBRATING  BODIES 


187 


So,  if  either  Fi  or  F"2  is  known,  the  other  may  be  at  once 
calculated  from  the  measurements  L\  and  Z2.  Thus,  assum- 
ing that  the  velocity  of  sound  in  air  is  known,  that  of  any 
solid  may  be  at  once  determined  by  making  a  rod  out  of  it, 
and  using  it  as  described.  The  velocity  in  any  other  gas 
or  in  any  liquid  may  also  be  determined  by  first  finding 
the  velocity  in  any  solid  rod  and  then  replacing  the  air  in 
the  tube  by  the  other  gas  or  the  liquid. 

This  method  is  not  so  accurate  as  is  desirable ;  but  it  is 
by  far  the  easiest  and  simplest  one  available  for  general 


use. 


TABLE   II 
VELOCITY  OF  SOUND 


Air     .... 

0° 

33,250 

cm   per  sec 

Hydrogen    .... 
Illuminating  Gas 
Oxvsren  . 

.     0° 
.    0° 
.     0° 

128,600 
49,040 
31,720 

tt         tt 
tt           a 

(C                ti 

Alcohol  (absolute)    . 
Petroleum    . 

.     8°.4 
.     7°  4 

126,400 
139  500 

tt            tt 

Water     . 

4° 

140  000 

tt            tt 

Brass  

361,700 

it            a 

Copper 

397,000 

tt            n 

Glass  

506  000 

tt            it 

Iron     .... 

509  300 

tt            tt 

Paraffin    .... 

16° 

130  400 

a           tt 

CHAPTEK  VII 

HARMONY  AND  MUSIC 

159.  Musical  Sounds.     As  already  stated  in  Article  124, 
the  simplest  musical  note  that  the  ear  can  recognize  is  that 
produced  by  a  harmonic  vibration.    It  has  also  been  known 
for  at  least  two  thousand  years  that  there  were-  certain 
combinations  of  sounds  which  were  pleasant  to  hear.    One 
such  combination  is  called  the  "  octave/'  which  is  due  to 
the  simultaneous  production  of  two  vibrations  of  suitable 
frequencies.     Another  combination  consists  of  three  vibra- 
tions, and  is  called  the  "  major  triad." 

160.  Numerical  Relations.     It  was  a  matter  of  great  in- 
terest to  measure  the  frequencies  of  these  vibrations  which 
compose  the  pleasing  combinations   known  for  so  many 
years.     It  was  found  that  whenever  a  vibration  and  an- 
other of  twice  the  frequency  took  place  together,  an  octave 
was  heard.    Thus,  if  two  tuning-forks  of  frequencies  n  and 
2  n  are  sounded  simultaneously,  an  octave  is  heard ;  and 
the  second  vibration  is  often  called  the  octave  of  the  first. 

Again,  it  was  found  that  when  three  vibrations  whose 
frequencies  were  HI,  n2,  n3,  were  sounded  together,  a  major 
triad  was  heard,  only  if 

n^  :  n2  :  n3  —  4  :  5  :  6. 

Thus  three  tuning-forks  whose  frequencies  are  200,  250, 
300,  will  cause  a  major  triad. 

161.  Harmony  and  Discord.    The  reason  why  these  simple 
combinations    of  vibrations  produce  a  pleasing  sensation 
remained  unknown  until  it  was  given  by  Von  Helmholtz. 


161]  HARMONY  AND  MUSIC  189 

As  explained  in  previous  chapters,  any  vibration  is,  as  a  rule, 
always  accompanied  by  partial  vibrations  whose  frequen- 
cies in  all  simple  cases  are  twice,  three  times,  etc.,  that  of 
the  fundamental  vibration  itself ;  it  has  also  been  shown 
that  beats  are  very  unpleasant  to  the  ear,  and  that  the.  de- 
gree of  the  disagreeable  sensation  depends  upon  the  num- 
ber of  beats  per  second  and  the  pitch  of  the  individual 
sounds.  Thus,  a  vibration  whose  frequency  is  200  is 
accompanied  by  partials  whose  frequencies  are  400,  600, 
800,  etc.  Consequently,  if  two  vibrations,  oi;  frequencies 
200  and  400,  are  produced  simultaneously  on  different 
instruments,  the  following  frequencies  occur : 

200,  400,    600,  etc. 
400,  800,  1200,  etc. 

and  there  are  therefore  no  unpleasant  beats  between  any 
of  the  vibrations.  So  it  is  seen  why  an  octave,  such  as 
200  and  400  form,  is  in  harmony. 

But  if  vibrations  of  frequencies  205  and  400  are  sounded 
together,  the  following  frequencies  occur : 

205,410,  615,  820,  etc. 
400,  800,  1200,  etc.  • 

And  now  there  are  10  beats  per  second  between  the  two 
vibrations  410  and  400,  also  20  per  second  between  820 
and  800,  etc. ;  and  so  this  combination  is  unpleasant,  and 
is  said  to  be  dissonant. 

Again,  if  three  vibrations  forming  a  major  triad  are 
sounded  together,  e.-g.  200,  250,  300,  the  following  fre- 
quencies occur: 

200,400,600,  800,  etc. 
250,  500,  750,  1000,  etc. 
300,  600,  900,  1200,  etc. 

No  beats  arise  ;  and  so  the  harmony  of  the  combination  is 
explained.  If  one  of  the  fundamental  vibrations  was  205 


190  THEORY  OF  PHYSICS  [CH.  VII 

instead  of  200,  there  would  be  beats ;  and  the  resulting 
sound  would  be  dissonant. 

Further,  Yon  Helmholtz  proved  that  in  every  discord 
there  were  beats,  generally  between  the  partial  vibrations ; 
while  in  harmony  the  beats  are  almost  entirely  absent. 

162.  Musical  Scales.  The  pitches  of  the  sounds  which 
compose  an  octave  or  a  major  triad  are  too  far  apart  to 
allow  any  complicated  music  to  be  composed  entirely  of 
them ;  and  so  the  attempt  has  been  made  at  various  times 
to  introduce  other  sounds  into  music,  and  to  make  thus 
what  is  called  a  musical  "  scale."  The  simplest  scale 
proposed  is  one  built  up  of  major  triads  and  octaves;  in 
the  interval  of  an  octave  seven  sounds  are  taken,  whose 
pitches  are  proportional  to  the  first  seven  of  the  following 
series  of  numbers : 

24,  27,  30,  32,  36,  40,  45,  48,  54,  60,  etc. 

This  series  of  numbers  may  be  continued,  as  shown,  by 
taking  the  octaves  of  the  original  series  of  seven. 

It  is  seen  that          24  :  30  :  36  =  4  :  5  :  6, 

36  :  45  :  2  X  27  =  4  :  5  :  6, 

•         32  :  40  :  48  =  4:5  :  6, 

so  that  the  scale  is,  as  it  were,  composed  of  major  triads  ; 
and  it  is  called  the  "  diatonic  "  scale.  Of  course  any  pitch 
may  be  taken  as  the  starting-point,  and  from  it  the  pitches 
of  the  other  notes  may  be  calculated.  Thus  it  has  been 
proposed  to  take  as  the  standard  a  note  whose  pitch  is  256  ; 
then  the  pitches  of  the  other  notes  in  the  scale  are  found 
by  multiplying  each  of  the  numbers  24,  27,  30,  etc.,  by 
256  /  24  or  32  /  3.  In  a  similar  manner  the  scale  between 
256  and  the  octave  below  it,  128,  can  be  built  up;  and 
thus  starting  from  any  arbitrary  pitch  the  scale  both  below 
and  above  it  may  be  constructed.  The  standard  pitch 
which  is  accepted  in  most  countries  to-day  is  261,  although 
this  recognition  is  by  no  means  universal. 


162]  HARMONY  AND  MUSIC  191 

The  "  interval "  between  two  notes  is  the  ratio  of  their 
pitches ;  and  it  is  at  once  seen  that  the  intervals  in  the 
diatonic  scale  are  most  uneven.  The  successive  intervals 
are  9  /  8,  10  /  9,  1 V  15,  9  /  8,  10  /  9,  9  /  8,  16  / 15,  etc.  An 
interval  9  /  8  or  10  / 9  is  called  a  "tone ; "  that  of  16  / 15,  a 
"  semi-tone."  An  attempt  was  made  to  equalize  these  in- 
tervals by  introducing  into  each  interval  of  a  tone  another 
note,  which  was  called  the  "  sharp  "  of  the  note  before  it, 
or  the  "  flat "  of  the  following  note,  and  thus  five  notes 
were  added. 

But  still  the  intervals  were  unequal,  and  so  a  radical 
change  was  made.  There  were  introduced  into  the  inter- 
val of  an  octave  twelve  notes,  the  intervals  between  suc- 
cessive ones  being  the  same.  Thus,  if  the  standard  note 
has  the  pitch  n,  and  if  the  interval  common  to  all  the 
notes  is  a,  the  scale  will  be  .  .  .,  n,  an,  a?n,  asn,  .  .  . 
alln,  a12  7i,  .  .  .  etc.  But  since  there  are  twelve  notes  in 
an  octave,  aVL  n  must  equal  2  n,  or  a  =  l^/2.  This  is 
called  the  "  tempered  "  or  the  "  chromatic  "  scale.  In  this 
scale  the  notes  no  longer  form  exact  major  triads  ;  and  so 
there  is  a  faint  discord  in  all  music  played  on  it ;  but  our 
ears  are  so  accustomed  to  it  that  it  is  rarely  noticed. 


BOOK    III 
HEAT 


UNIVERS/TY  OF  CALIFORNIA 


QF  PHYSICS 

BOOK  III 


HEAT 


INTRODUCTION 

EVERY  one  is  familiar  with  the  sensations  of  heat  and 
cold ;  and  it  is  well  known  that  the  causes  which  produce 
these  sensations  can  also  produce  other  effects.  The  study 
of  the  nature  of  these  so-called  "  heat-effects  "  and  of  the 
laws  according  to  which  they  are  produced  forms  what 
is  known  in  Physics  as  the  subject  of  "  Heat." 

163.  It  was  noted,  in  speaking  of  Work  and  Energy 
(Art.  67),  that,  whenever  work  is  done  against  fric- 
tion, heat-effects  are  produced.  Ice  may  be  melted  by 
friction ;  water  may  be  heated  until  it  boils  by  friction ; 
a  piece  of  wood  may  be  set  on  fire  by  friction,  if  suitably 
applied.  The  amount  of  the  heat-effect  may  be  measured 
by  the  amount  of  the  ice  melted,  the  amount  of  the  water 
boiled,  etc. ;  and  it  is  found,  as  the  result  of  most  careful 
experiments,  that  the  amount  of  heat-effect  depends  only 
upon  the  amount  of  work  done  against  friction. 

Of  course  heat-effects  may  be  produced  in  other  ways 
than  by  friction.  If  a  gas  is  compressed,  its  temperature 
is  raised ;  and  this  is  a  well-known  heat-effect.  But  here, 
too,  work  is  required  to  compress  the  gas ;  and  the  amount, 
of  the  heat-effect  produced  depends  only  upon  the  amount 
of  work  done.  When  a  substance  is  burnt,  there  are 
heat-effects.  For  instance,  if  carbon  combines  chemically 
with  oxygen,  as  it  does  in  an  ordinary  fire,  there  are  heat- 
effects  ;  and  their  amount  depends  only  upon  the  loss  of 
potential  energy  of  the  carbon  and  oxygen  which  combine. 


196  THEORY  OF  PHYSICS 

So,  in  general,  it  may  be  proved  that  the  numerical  value 
of  the  heat-effect,  whatever  it  is,  depends  only  upon  the 
work  done ;  or,  what  is  the  same  thing,  upon  the  loss  of 
energy  by  the  bodies  doing  the  work. 

It  may,  therefore,  be  stated  that  heat-effects  are  manifes- 
tations of  energy ;  because,  in  their  production,  energy  has 
passed  from  certain  bodies  into  those  in  which  the  "  heat- 
effects  "  occur.  Further,  it  is  noticed  that  in  every  heat- 
effect  the  work  done  is  against  "  forces  "  which  act  between 
most  minute  portions  of  matter  or  is  concerned  in  some 
way  with  the  motion  of  these  minute  portions.  This  is 
equivalent  to  saying  that,  in  the  production  of  heat-effects, 
energy  becomes  associated,  in  either  a  potential  or  kinetic 
form,  with  portions  of  matter  which  are  of  the  general 
dimensions  of  molecules.  In  an  exactly  similar  manner, 
it  may  be  stated  that  if  energy  is  taken  away  from  the 
molecules  there  is  what  may  be  called  a  negative  heat- 
effect,  such  as  fall  in  temperature,  contraction,  etc.  These 
facts  will  be  shown  more  in  detail  in  Chapter  I. 

Since,  then,  all  heat-effects  are  caused  by  the  passage  of 
energy  into  an  association  with  minute  portions  of  matter, 
the  amount  of  any  heat-effect  must  be  measured  by  the 
amount  of  energy  which  is  involved  in  the  phenomenon. 
That  is,  all  heat-effects  must  be  measured  in  ergs  (Art.  68). 

164.  The  most  important  sources  of  heat-energy  are  the 
following :  — 

a.    Such  mechanical  actions  as  friction,  compression,  etc. 

I.  Many  chemical  actions,  such  as  combustion,  especially 
of  carbon  and  oxygen. 

c.  The  sun,  which  directly   or   indirectly  is  the    chief 
source  of  the  energy  available  for  our  use. 

d.  Electric  currents,  because,  whenever  a  current  passes, 
the  temperature  of  the  conductor  is  raised. 

e.  Changes   in  molecular   arrangement,  which   will  be 
more  fully  discussed  in  Chapter  IV.,  e.  g.  freezing  of  water. 

/.  Hot-springs,  volcanoes,  etc.,  all  being  evidences  of  the 
high  temperature  of  the  interior  of  the  earth. 


CHAPTER  I 

HEAT-EFFECTS  —  TEMPERATURE 

AMONG  the  most  familiar  heat-effects  are  the  following : 

1.  Change  in  volume. 

2.  Change  in  temperature. 

3.  Change  in  molecular  arrangement  or  structure,  such 
as  melting,  boiling,  etc. 

There  are  also  electrical  and  chemical  effects ;  but  these 
three  are  the  most  important. 

165.  Change  in  Volume.  It  is  well  known  that,  if  any 
body  is  warmed  over  a  fire  or  in  other  ways,  its  volume  is 
changed.  It  in  general  expands.  In  this  act  of  expansion, 
work  is  done  in  at  least  two  ways  :  first,  the  molecules  of 
the  body  are  forced  farther  apart;  second,  whatever  is 
resting  on  the  body  is  moved.  This  first  change,  that  of 
position  of  the  molecules,  evidently  belongs  to  the  third 
effect  noted  above,  "  Change  in  molecular  arrangement ; " 
and  it  will  be  discussed  later.  The  work  done  by  the 
expanding  body  against  the  matter  touching  it  is  some- 
times called  "external  work."  Thus,  if  a  pillar  which 
supports  a  building  expands,  it  raises  the  building,  and  so 
does  work.  An  ordinary  solid  which  is  surrounded  by  air 
does  work  in  expanding  by  raising  the  atmosphere  around . 
it.  So  it  may  be  said  that,  in  expansion,  energy  has  left 
the  body  producing  the  expansion;  and  this  has  been 
spent  in  two  ways  at  least,  —  in  overcoming  some  external 
force  and  in  producing  molecular  re-arrangements.  It  is 
possible,  of  course,  that  these  molecular  re-arrange  men ts 
may  be  accompanied  by  a  decrease  in  the  potential  or 
t 


198  THEORY  OF  PHYSICS  [CH.  I 

kinetic  energy  of  the  molecules ;  and  in  this  case,  instead 
of  work  being  required  to  produce  the  molecular  change, 
this  change  itself  would  furnish  energy,  which  would  be 
available  for  external  work. 

If  the  body  contracts,  there  are,  of  course,  molecular 
changes  ;  the  external  forces  now  themselves  do  work  ;  and 
so  the  body  receives  energy. 

In  any  case,  if  there  is  a  uniform  pressure,  p,  over 
the  entire  body,  and  if  its  volume  changes  from  VQ  to  V1} 
the  external  work  done  is  p  (Vi  —  VQ]  (see  Art.  95).  If 
VQ  <  Vi,  this  work  is  done  against  the  external  forces.  If 
F"0  >  Fi,  the  work  is  negative,  meaning  that  it  is  done  by 
the  forces ;  and  energy  is  given  the  body. 

166.  Change  in  Temperature.  That  heat-effect  which  is 
best  known  is  change  in  temperature,  because  our  senses 
allow  us  to  tell  with  some  degree  of  accuracy  that  when 
held  over  a  fire,  a  body,  as  a  rule,  becomes  hotter.  All 
bodies  do  not,  however,  become  hotter  under  these  condi- 
tions. For  instance,  a  block  of  ice  may  be  heated  until  it 
all  melts  ;  and  there  is  no  change  in  temperature.  But 
these  questions  will  be  discussed  later.  So  will  the 
more  exact  ideas  of  temperature,  and  the  methods  of 
measurement. 

As  was  stated  in  the  general  introduction  (Art.  5), 
the  present  theory  of  matter  is  to  regard  all  minute  por- 
tions of  it  as  being  in  motion.  We  shall  see  evidences  of 
this  later  on ;  and  it  will  be  shown  also  that  in  gases  there 
is  no  question  but  that  the  temperature  and  the  average 
kinetic  energy  of  translation  of  the  molecules  are  most 
intimately  connected.  If  this  energy  is  increased  in  any 
way,  there  is  an  increase  in  the  temperature,  and  vice- 
versa.  It  is  also  believed  that  this  is  more  or  less  true  of 
solids  and  liquids.  So  that,  if  the  work  done  on  the  body 
is  spent  in  increasing  the  kinetic  energy  of  translation  of 
the  molecules,  the  particular  heat-effect  manifested  is  rise 
in  temperature. 


167]  HEAT-EFFECTS  —  TEMPERATURE  199 

167.  Molecular  Changes.  Any  alteration  of  molecular 
arrangement  or  structure  must  involve  changes  in  energy, 
either  potential  or  kinetic.  Changes  in  kinetic  energy  are 
in  general,  as  just  noted,  made  evident  by  changes  in  tem- 
perature. But  among  so-called  heat-effects  there  are  a 
great  many  changes  which  are  mainly  changes  in  potential 
energy  of  the  molecular  structures.  Fusion,  or  the  change 
of  state  of  a  body  from  solid  to  liquid,  is  an  illustration  of 
this.  In  this  process  work  must  be  done  on  the  molecules, 
that  is,  energy  must  be  added  to  them,  so  as  to  give  them 
their  increased  freedom  of  motion,  which  they  have  in  the 
liquid  state.  Conversely,  if  a  liquid  is  solidified,  energy 
must  be  taken  away  from  the  molecules  ;  and  the  amount 
taken  away  must  equal  exactly  the  amount  required  to 
liquefy  the  solid. 

Evaporation,  or  the  change  of  state  of  a  body  from 
liquid  to  gas,  is  another  illustration.  So  is  sublimation,  or 
the  change  from  solid  to  gas  directly.  Any  change  in  vol- 
ume also  presupposes  molecular  changes ;  and  this,  in 
general,  involves  changes  in  potential  energy.  So  does 
change  in  shape.  Dissociation  is  also  an  example,  because 
in  this  phenomenon  molecules  are  broken  up  into  simpler 
parts.  Combination,  the  reverse  of  dissociation,  is  another. 
So  is  solution  of  one  body  in  another,  because  the  mole- 
cules (or  their  parts)  become  differently  arranged. 

In  all  these  cases,  if  there  is  an  increase  in  potential 
energy,  work  is  done  on  the  molecules.  This  gain  in  po- 
tential energy  may  come  from  the  fact  that  the  molecules 
themselves  are  losing  kinetic  energy,  and  in  this  case  the 
temperature  falls;  or,  if  the  temperature  remains  constant, 
the  energy  must  come  from  outside.  If  there  is  a  decrease 
in  potential  energy,  this  may  correspond  to  an  increase  in 
kinetic  energy  and  so  to  a  rise  in  temperature,  or  to  an 
emission  of  energy  to  surrounding  bodies. 

The  internal  changes  are,  then,  changes  in  temperature 
and  changes  in  molecular  arrangement  or  structure ;  and, 


200  THEORY  OF  PHYSICS  [CH.  1 

as  a  result  of  them,  the  molecular  energy  of  the  body  is 
changed.  The  molecular  energy  of  a  body  at  any  instant 
is  called  its  "  intrinsic  energy  "  in  that  condition  ;  and  so 
these  changes  are  called  changes  in  intrinsic  energy.  If 
the  body  ever  returns,  after  any  cycle  of  changes,  to  its 
original  condition,  its  intrinsic  energy  is  the  same  as  be- 
fore, because  it  depends  only  upon  the  body  itself. 

When  molecular  energy  is  given  a  body,  other  effects 
than  these  may,  of  course,  be  produced,  such  as  electrical 
effects ;  but  the  Conservation  of  Energy  requires  that  the 
energy  added  shall  exactly  equal '  the  sum  of  the  external 
and  the  internal  work  done.  This  is  perfectly  borne  out 
by  all  known  experimental  evidence.  Stated  in  other 
words :  energy  added  =  change  in  intrinsic  energy  4- 
external  work  done. 

TEMPERATURE 

168.  The  temperature  of  a  body  is  a  property  of  the 
body  which  expresses  its  thermal  relation  to  other  bodies. 
One  body  is  said  to  have  a  higher  temperature  than  an- 
other, if,  when  the  two  are  placed  in  intimate  contact,  the 
first  loses  molecular  energy  and  the  second  gains  it.  If  a 
body  is  gaming  molecular  energy,  some  heat-effect  is  being- 
produced  ;  and,  since  heat-effects  are  easily  observed,  it  may 
at  once  be  seen  which  body  has  the  higher  temperature.  If 
two  bodies  are  at  the  same  temperature,  neither  one  gains 
or  loses  molecular  energy ;  and  this  condition  may  be  easily 
determined  by  experiment.  Further,  it  is  proved  by  experi- 
ment that,  if  two  bodies  have  the  same  temperature  as  a 
third  body,  the  two  bodies  themselves  have  the  same  tem- 
perature. This  fact  permits  the  comparison  of  the  tempera- 
tures of  two  bodies  which  cannot  be  conveniently  placed  in 
contact,  by  means  of  a  third  body  which  can  be  compared 
with  the  two  separately. 

This  third  body  must  have  some  property  which  is 
easily  affected  by  changes  in  molecular  energy  ;  but  it  is 


168]  HEAT-EFFECTS  -TEMPERATURE  201 

entirely  immaterial  what  property  this  is,  only  it  must 
admit  of  exact  measurement.  Such  a  body  may  be  called 
a  "  thermometer,"  because  by  means  of  it  a  numerical  value 
may  be  given  to  the  temperature  of  any  body. 

In  the  ordinary  thermometer,  changes  in  temperature 
are  recognized  by  changes  in  volume  of  some  fluid,  e.g. 
mercury  or  alcohol  or  air,  enclosed  in  glass.  The  altera- 
tions in  volume  are  registered  by  divisions  made  on  the 
glass  tubes  which  enclose  the  fluid ;  a  liquid  has  of  course 
a  free  surface,  but  a  gas  may  be  separated  from  the  out- 
side air  by  a  short  mercury  index  which  marks  the  vol- 
ume. The  volume  of  the  fluid  may  be  noted  under  two 
different  definite  conditions.  Call  these  two  volumes  vl 
and  vz.  Let  it  be  desired  to  distinguish  between  the  two 
given  conditions  n  degrees  of  temperature.  Then  one 
degree  may  be  defined  as  corresponding  to  a  difference  in 
volume  of  the  fluid  of  (v2  —  vi)  /  n.  So,  if  t\  is  a  number 
given  arbitrarily  to  the  temperature  which  corresponds  to 
the  volume  vi,  the  numerical  value  of  the  temperature 
which  corresponds  to  the  volume  v  is 

t  =  tl  +  *-=*  =  tl  +  nV-^l  (1) 


If  mercury  is  the  fluid  used,  the  thermometer  is  called  a 
"  mercury-thermometer."  If  air  is  used,  the  instrument  is 
called  an  "  air-thermometer,"  etc.  By  means  of  such  an 
instrument,  a  numerical  value  can  be  given  to  the  tempera- 
ture of  any  body  ;  but  the  number  depends  upon  two  arbi- 
trary quantities,  ti  and  n,  and  upon  the  fluid  used,  and  so 
is  itself  perfectly  arbitrary. 

By  general  consent  the  two  definite  conditions  which  fix 
the  two  standard  temperatures  ti  and  U  are  the  tempera- 
ture of  equilibrium  of  ice  and  water  and  of  water  and 
steam,  when  the  corrected  atmospheric  pressure  is  76  cm. 

of  mercury.     (See  Art.  175.)     Accurate  experiments  have 

7* 


202  THEORY   OF  PHYSICS  [CH.  I 

shown  that  these  two  temperatures  are  always  the  same 
the  world  over ;  and  they  are  also  very  convenient  for  ordi- 
nary purposes.  Further,  by  universal  agreement  among 
scientists,  there  are  distinguished  100  intermediate  degrees 
of  temperature ;  and  the  temperature  of  equilibrium  of  ice 
and  water  is  called  0°  ;  i.e.n  =  100,  ti  =  0.  If  VQ  is  the 
volume  at  0°  and  vwo  that  at  100°,  the  temperature  which 
corresponds  to  any  volume,  v,  is 

t  =  0°  +  100°  v  ~  VQ     ,  (2) 

Vioo  —  VQ 

This  number  is  called  the  temperature  on  the  "  centi- 
grade "  scale  with  the  mercury-,  or  air-,  etc.,  thermometer. 
This  scale  can  obviously  extend  below  0°  and 
yy  above  100°,  if  v  is  less  than  v0  or  greater  than 

VIQQ. 

A  mercury-thermometer  consists,  in  general, 
of  a  glass  bulb  with  a  long  narrow  glass  tube 
attached,  the  mercury  filling  the  bulb  and  part  of 
the  tube.  This  tube  is  closed  at  the  further  end ; 
and  the  space  above  the  mercury  is  often  a  vac- 
uum, although  sometimes  a  gas  like  nitrogen  is 
put  in  under  low  pressure.  The  narrow  tube  is 
supposed  to  be  perfectly  uniform ;  and  divisions 
are  marked  on  it  corresponding  to  equal  changes 
in  length.  The  instrument-maker  generally  marks 
0  at  the  point  where  the  mercury  stands  at  0°, 
and  100  where  it  stands  at  100°,  dividing  the 
intervening  column  into  100  equal  parts.  Of 
course,  any  maker  is  liable  to  commit  an  error, 
and  so  each  thermometer  must  be  carefully  stand- 
ardized and  calibrated.  That  is,  it  must  be  tested 
to  see  if,  when  at  the  temperature  of  equilibrium 
FIG.  132.  °f  i°e  and  water,  the  reading  on  the  scale  is  0  ; 
and  if,  when  at  the  temperature  of  equilibrium 
of  water  and  steam  under  the  pressure  of  the  standard 


168]  HEAT-EFFECTS  —  TEMPERATURE  203 

atmosphere,  the  reading  is  100.  And  further,  it  must  be 
examined  to  see  if  the  long  tube  is  perfectly  uniform,  so 
that  equal  lengths  correspond  to  equal  volumes.  Methods 
for  this  examination  are  taught  in  all  laboratories.  (It 
may  be  noted  here,  simply  for  reference,  that  if  the  at- 
mospheric pressure  is  1  cm.  greater  than  76  cm.  of  mer- 
cury, the  temperature  of  equilibrium  of  water  and  steam 
rises  0.°366  centigrade;  and  for  other  small  variations  in 
pressure  the  changes  in  temperature  are  in  proportion.  If 
the  pressure  falls,  the  equilibrium-temperature  also  de- 
creases. See  Article  194.) 

An  ordinary  thermometer  is  liable  to  many  errors,  all  of 
which  must  be  carefully  taken  into  account.  These  neces- 
sary corrections  are  all  explained  in  laboratory  manuals, 
and  are,  in  almost  all  cases,  due  to  the  fact  that  glass  is  a 
substance  which,  when  once  expanded,  does  not  return  to 
its  previous  volume  after  the  temperature  is  brought  back 
to  its  former  value,  until  many  days  or  months  have 
elapsed. 

A  mercury-thermometer  cannot  be  used,  of  course,  ex- 
cept between  those  temperatures  for  which  mercury  re- 
mains a  liquid,  that  is,  between  —  39°  C.  and  350°  C.  And, 
since  mercury  expands  at  different  rates  at  different  tem- 
peratures, there  is  no  agreement  between  the  meaning  of 
1°  at  different  temperatures.  A  gas-thermometer  is  free 
from  these  limitations,  because  it  does  not  liquefy  except 
at  enormously  low  temperatures,  and  because  it  expands 
nearly  uniformly  under  constant  conditions  of  pressure. 
But  a  gas  must  be  enclosed  in  some  solid ;  and,  as  this 
melts  at  some  high  temperature,  even  a  gas-thermometer 
cannot  be  used  at  very  high  temperatures.  A  gas-ther- 
mometer is  really  used  but  rarely,  except  for  standardizing 
purposes,  owing  to  the  large  volume  which  the  instrument 
must  necessarily  have. 


CHAPTEE  II 

CHANGES  IN  VOLUME 

SOLIDS 

169.  Linear  Expansion.  Whenever  a  solid  is  held  over  a 
fire,  or  is  otherwise  heated,  its  volume  and  its  temperature 
both,  as  a  rule,  change.  (If  the  solid  is  at  its  melting 
temperature,  this  is  not  so.)  If  the  solid  is  isotropic,  i.  e. 
has  the  same  properties  in  all  directions,  the  expansions 
of  all  lines  of  equal  length  in  the  body  are  the  same,  as 
may  be  proved  by  experiment.  If  the  body  is  not  iso- 
tropic, e.  g.  if  it  is  a  crystal,  this  is  not  true. 

If  a  long  rod  of  any  substance  has  its  length  measured 
at  different  temperatures,  there  is  a  connection  between 
the  change  in  length  and  the  change  in  temperature.  Thus 

let 

ai  =  length  at  temperature  £b 
n   —     it         <(  ii  f  . 

«2   —  C2  j 

then  it  is  found  by  experiment  that 

«2  —  #1  =  a  «i  (t2  —  ti) (1) 

where  a  is  very  nearly  a  constant  for  any  one  substance 
for  different  values  of  tz. 

If  t2  —  ti  =  1,  i.  e.  if  the  temperature  is  raised  one  degree, 

a  —  J! 5:  j  and    this  ratio  is  called  the  "  coefficient  of 

ai 

linear  expansion  "  of  this  substance  at  the  temperature  t\. 
a  is  nearly  the  same  at  all  temperatures,  but  not  exactly. 

If  tz  is  not  equal  to  ti  +  1,  the  ratio  a  —  — -1  .  —  is 

tz  —  ti      ai 


169] 


CHANGES  IN  VOLUME 


205 


called  the  "  average  coefficient  of  linear  expansion  between 
t\  and  tz" 

The  figures  in  the  following  table  refer  to  the  changes 
in  length  of  rods  of  different  substances,  which  are  first 
measured  at  0°  C. ;  i.  e.  ^  =  0,  and  cti  =  a0  is  the  length 

at  0°, 

—  «o. 


a  = 


=  a0(l  +  afa 
TABLE  III 


(2) 


AVERAGE  COEFFICIENTS  OF  LINEAR  EXPANSION  BETWEEN 
0°  AND  100°  C. 


Aluminum  .... 

0.000023 
0.000018 

Platinum    .  . 
Silver    .... 

0.000009 
0.000019 

Copper 

0  000017 

Steel  

0.000011 

Glass      

0.000009 

Tin  

0.000023 

Iron  (soft) 

0.000012 

0.000029 

Iron  (cast)  .... 

0.0000105 

Since  a  is  different  for  different  substances,  it  is  possible 
so  to  combine  different  rods  that  the  resulting  length  does 
not  change  when  the  temperature  is  altered.  This  is  the 
principle  of  one  form  of  compensating  pendulum. 

If  the  body  is  isotropic,  as  stated  above,  a  is  the  same 
for  all  directions  ;  but  if  the  body  is  not  isotropic,  a  has 
different  values  in  different  directions.  In  a  crystal  which 
has  three  different  axes  at  right  angles  to  one  another,  a 
has  three  different  values  corresponding  to  these  three  direc- 
tions. In  general  a  is  positive,  i.  e.  the  length  of  a  line 
of  the  body  increases  as  the  temperature  rises ;  but  this  is 
not  true  in  all  cases  :  for  instance,  if  the  temperature  of 
a  stfetched  rubber  pord  is  raised,  it  tends  to  contract. 

The  measurement  of  a  for  a  solid  is  not  difficult,  as  it 


206  THEORY  OF  PHYSICS  [CH.  II 

involves  only  the  measurement  of  the  length  of  a  bar  at 
two  different  temperatures.  Fuller  details  are  given  in 
laboratory  manuals.  (The  most  accurate  method  depends 
upon  the  interference  of  light- waves,  and  the  principle  will 
be  discussed  later.) 

170.  Application  of  Principle  of  Stable  Equilibrium.     If  a 
bar  expands  when  its  temperature  is  raised,  stretching  the  bar 
mechanically  will  cool  it.     For  a  bar  held  in  a  region  of 
constant  temperature  is  in  stable  equilibrium ;  and  in  ac- 
cordance with   the   principle   of   stable   equilibrium   (see 
Art.  66),  if  when  the  temperature  rises  expansion  is  pro- 
duced, expansion  of  itself  must  tend  to  cause  a  fall  in 
temperature.     Compression  will  produce  a  rise  in  tempera- 
ture. 

If,  on  the  other  hand,  the  substance  of  the  bar  is  such 
that  raising  the  temperature  produces  contraction,  stretch- 
ing will  produce  a  rise  in  temperature. 

171.  Cubical   Expansion.      Since    each    line    in    a    solid 
changes  when  its  temperature  is  altered,  the  volume  will 
also  change  (unless  the  body  is  crystalline  and  one  axis 
contracts  or  expands  enough  to  balance  the  change  of  the 
other  two).    In  an  isotropic  body  whose  coefficient  of  linear 
expansion,  referred  to  0°  C.,  is  a,  formula  (2)  gives : 


a  •= 


or  «2  =  a0  (1  +  a  tz). 

Consider  a  cube  each  of  whose  edges  had  the  length  a0 
at  0°  C.  Then  at  the  temperature  £2°,  each  edge  will  have 
the  length  a2.  The  initial  volume,  VQ,  was  a8 ;  the  final 
volume,  v2,  at  t2°  is  a23.  Hence,  cubing  formula  (2), 

V2  =  V0  (1  +  a  t2)3. 

But  a  is  so  small  for  all  solids  that  a2  is  too  minute  to 
notice.  Hence  • 


172]  CHANGES   IN  VOLUME  207 

Or  v2  =  v0  (1  +  3  a  t2). 

That  is,  3  a  =  ^2  ~V° . 

And  so,  corresponding  to  the  coefficient  of  linear  expan- 
sion a,  the  coefficient  of  cubical  expansion  is  3  a.  This  is 
sometimes  written  /3.  That  is, 


(3) 

/  -4  S*k        „        \  I 

or 

A  solid  which  has  in  it  any  kind  of  a  cavity  expands  ex- 
actly as  if  there  were  none.  For  imagine  the  actual  solid 
to  be  constructed  of  minute  cubes  (or  bricks).  As  the  tem- 
perature is  raised,  each  cube  expands  into  a  larger  cube ; 
and  the  surface  of  any  cavity  bounded  by  these  larger  cubes 
is  just  as  much  larger  than  it  was  before  as  it  would  have 
been  if  the  space  had  been  solid  and  not  hollow. 

LIQUIDS 

172.  Cubical  Expansion.  If  the  temperature  of  a  liquid  is 
changed,  its  volume  is  also ;  and  the  connection  between 
these  changes  may  be  expressed  as  it  was  for  a  solid.  If 
v1  and  v2  are  the  volumes  of  a  particular  liquid  at  the  tem- 
peratures ti  and  t2,  it  is  found  by  experiment  that 

v2  -  Vi  =  j3  vl  (t2  -  h), (4) 

where,  for  a  given  liquid,  /3  is  nearly  a  constant  for  various 
values  of  t2,  and  is  called  the  average  coefficient  of  cubical 
expansion  between  the  temperatures  ti  and  t2.  As  gener- 
ally measured,  ft  refers  to  the  centigrade  scale  and  to  the 
temperature  0°  as  the  starting-point.  That  is,  v\  =  VQ,  t1  =  0. 

Hence,  v2  —  v0  = 

or  v2  = 

So  3  = 


208  THEORY   OF  PHYSICS  [CH.  II 

Some  values  of  /3  for  different  substances  are  given  in 
table.  It  will  be  noticed  that  they  are  larger  than  for  the 
ordinary  solids. 

TABLE   IV 
AVERAGE  COEFFICIENTS  OF  CUBICAL  EXPANSION  OF  LIQUIDS 


Alcohol  . 
Ethyl  Ether 

0°-80°  0.00105 
0°-33°  0.00210 

Mercury    . 
Turpentine 

0°-100°  0.000182 
9°-106°  0.00105 

173.  Water,     fi  is  not  constant  for  any  one  liquid,  but 
changes  with  the  temperature,  just  as  it  does  with  solids. 
In  one  liquid,  water,  /3  changes  greatly.     In  fact,  between 
0°   and  4°  C.     (3  for  water  is  negative ;  while  from  4°  to 
100°  C.  it  is  positive.     So  water  is  densest  at  4°  C.  as  noted 
in  Article  8.     This  fact  is  most  important  in  the  econ- 
omy of  nature  ;  because,  as  the  water  in  a  lake  or  river  be- 
comes cold,   the  denser  portions  sink  to  the  bottom ;  and 
so,  while  the  water  on  the  top  may  be  0°,  that  at  the  bot- 
tom will  be  only  4°  C.     Consequently,    lakes  and   rivers 
freeze  on  the  top,  not  on  the  bottom. 

174.  Application  of  Principle   of   Stable   Equilibrium.     A 
liquid  kept  in  a  region  of  constant  temperature  is  in  stable 
equilibrium.     So  if,  when  the  temperature  of  the  liquid  is 
increased,  there  is  expansion,  then  expanding  the  liquid 
will  produce  a  fall  in  temperature,  and  compressing  the 
liquid  will  cause  a  rise  in  temperature.    But,  if  the  liquid  is 
one  which  contracts  when  its  temperature  is  raised,  then 
compressing  it  will  produce  a  fall  in  temperature. 

175.  Barometric  Correction,     As  explained  in  the  section 
on  the  mercury  barometer  (Art.  100),  the  actual  height  of 
the  mercury  column  varies  with  the  temperature,  because 
the  density  of  the  mercury  changes ;  and  so  the  observed 
height   is    "  corrected  "    to  give  the  height   to  which  the 
column  would  have  risen  if  the  density  of  the  mercury  had 


175]  CHANGES  IN  VOLUME  209 

been  that  which  it  is  at  0°  C.  This  correction  is  made 
in  this  way.  Let  p  and  h  be  the  density  and  height,  actu- 
ally observed  at  t°  C  ;  and  pQ  and  h0  the  density  at  0°  and 
the  "  corrected  "  height.  p0  and  h0  must  be  such  that  the 
pressure  due  to  a  column  of  height  h0  and  density  /30  shall 
equal  the  existing  pressure.  That  is, 

pgh  =  p0gh0, 

7,  h 

or  ho  =  p  —  • 

po 

But  the  change  in  density  of  1  gram  of  mercury  when  its 
temperature  is  changed  from  0°  to  t°  C.  is  easily  calculated. 
Call  vQ  the  volume  at  0°  ;  and  v  that  at  t°  C.  Hence 

1  =  v0pQ  =  vp.     .'.   P/PO  =  VO/V    ...     (6) 
But  v  =  vl  ' 


where  /3  is  the  average  coefficient  of  cubical  expansion  of 
mercury  between  0°  and  t°  C.     Hence 


andso  fc0  =  _-  =  fc(i_£Q,       ...     (8) 


since  ft  is  so  small  that  its  square  may  be  neglected  in 
comparison  with  1. 

But  h  is  in  general  measured  in  divisions  ruled  on  the 
metal  or  glass  case  which  encloses  the  mercury;  and,  if 
one  of  these  divisions  is  1  cm.  long  at  0°  C.  its  length  in 
cm.  at  t°  is  a  =  a0  (1  +  cut),  where  a0  =  1,  and  a  is  the 
average  coefficient  of  linear  expansion  of  the  metal  or 
glass.  That  is, 

a  —  1  -f  a  t. 

Hence  the  true  corrected  height  in  centimetres  is 

fc0  =  A(l  +  aO(l-£0  =  fc[l-(£-a)*].     •     (9) 
For  mercury  /3  =  0.0001816  ; 

For  brass  a  =  0.000018. 


210 


THEORY  OF  PHYSICS 


[CH.  II 


And  so  for  an  ordinary  mercury  barometer  with  a  brass 
case  . 

A0  =  fc(l -0.000160 (10) 

176.  Measurement  of  /&  A  liquid  must  obviously  be 
held  in  a  solid;  and  so,  when  the  temperature  is  changed, 
the  volume  of  the  solid  changes  as  well  as  that 
of  the  liquid.  Consequently,  the  apparent  or 
the  observed  change  of  the  liquid  is  the  actual 
true  change  less  the  change  of  the  solid.  If  a 
liquid  is  enclosed  in  a  bulb  with  a  fine  con- 
nected tube,  the  volumes  of  equal  lengths  of 
which  are  known,  the  apparent  change  may  be 
at  once  measured  by  the  change  in  height  of  the 
top  of  the  liquid  column.  So,  if  the  volume  of 
the  bulb  is  known,  and  if  it  is  of  a  substance 
whose  coefficient  of  cubical  expansion  is  known, 
its  total  expansion  may  be  calculated ;  and  then 
the  true  expansion  of  the  liquid  is  at  once  given. 
From  a  knowledge  of  this  true  expansion,  the 
change  in  temperature  and  the  initial  volume  of 
the  liquid,  0  may  be  calculated,  because 


FIG.  133.  This  same  experiment  permits  one  to  measure 
the  cubical  coefficient  of  expansion  of  the  solid 
which  encloses  the  liquid,  if  the  absolute  expansion  of  the 
liquid  is  known.  This  method  is  very  commonly  used, 
because  the  absolute  expansion  of  mercury  is  quite  accu- 
rately known. 

To  avoid  the  necessity  of  determining  the  coefficient 
of  expansion  of  the  enclosing  solid,  another  method  has 
been  devised,  which  is  independent  of  the  nature  of  the 
solid.  Its  principle  depends  upon  the  fact  that  the  height 
to  which  a  liquid  rises  above  its  free  surface  in  a  tube 


176] 


CHANGES  IN  VOLUME 


211 


which  dips  into  a  basin  of  the  liquid  is  entirely  inde- 
pendent of  the  size  or  shape  of  the  tube,  and  depends 
only  upon  the  pressure  and  the  den- 
sity. A  double  U-tube  is  constructed 
with  the  outer  open  arms  quite  long. 
The  liquid  to  be  examined  is  poured 
into  the  outer  arms,  but  the  two  por- 
tions of  the  liquid  are  not  allowed  to 
meet  in  the  middle  branch.  One-half 
the  tube  is  enclosed  in  a  space  whose 
temperature  is  ti\  and  the  other  in  a  . 
space  whose  temperature  is  t%.  By 
compressing  the  air  in  the  middle  por- 
tion of  the  tube,  the  liquid  may  be 
forced  up  to  considerable  heights  in 
the  two  outer  arms.  Let  the  two  ver- 
tical heights  of  the  surfaces  of  the 
outer  columns  above  the  free  surfaces 
of  their  respective  inner  columns  be 
hi  and  h2,  and  let  their  respective  densities  be  pi  and  pz. 
Then,  since  both  columns  have  the  same  pressures  at  both 
their  outer  and  their  inner  ends, 

pig  hi  =  pzghz, 
or  pi  hi  =  pz  hz. 

But,  considering  the  volumes  vl  and  vz  of  one  gram  of  the 
liquid  at  the  temperatures  ti  and  U 

1  =  pi  Vi  = 
Hence  vz  hi  =  Vi  hz. 


FIG.  134. 


But 


Hence 


—  hi 


—  ti  hi 


All  the  quantities  in  the  fraction  can  be  measured,  and  so 
8  may  be  determined. 


212  THEORY  OF  PHYSICS  [CH.  II 

GASES 

177.  Cubical  Expansion,     All  gases  expand  as  the  tem- 
perature rises  if  the  pressure  is  kept  constant,  and  the 
connection  between  volume  and  temperature  may  be  ex- 
pressed by  the  equation 

v  =  v0(l  +  ftt),      .    .     .    .'.  .     (12) 

where  v0  is  the  volume  at  0°  C.,  v  is  the  volume  at  t°  C., 
and  ft  is  the  average  coefficient  of  cubical  expansion  at 
constant  pressure,  referred  to  0°  C. 

ft  is  not  alone  nearly  constant  for  any  one  gas  ;  but  it  is 
also  nearly  the  same  for  all  gases.  This  is  a  most  remark- 
able fact,  and  bears  witness  to  the  simplicity  of  the  laws 
of  gases.  That  ft  is  the  same  for  all  gases  is  called  Charles' 
Law,  sometimes  Gay-Lussac's.  Its  value  is  almost  exactly 
0.003665. 

Since,  in  the  expansion  of  a  gas  at  constant  pressure,  the 
mass  of  the  gas  does  not  change,  the  above  formula  may 
be  changed  into 

pQ  =  p(l  +  ft  0, 

,  mass  mass 

because  v0  =  -  ,  v  =  -  , 

Po  p 

and  so  v0p0  =  v  p. 

178.  Change  of  Pressure  and  Temperature.    If  the  pressure 
changes,  the  volume  does  also  in  general.     In  particular, 
if  the  temperature  is  constant,  and  the  pressure  changes, 
Boyle's  Law  (see  Art.  108)  gives  the  connection  between 
the  pressures  and  volumes  (or  densities),  viz.  : 

p  v  =  k  m, 
or 


where  k  is  a  constant  for  any  gas  at  any  one  temperature. 
If  both  temperature  and  pressure  change,  the  process  may 
be  considered  as  taking  place  in  two  stages,  —  a  change  in 


178]  CHANGES   IN   VOLUME  213 

temperature  at  constant  pressure,  then  a  change  in  pres- 
sure at  constant  temperature. 

Let  the  gas  have  a  density  p0  at  0°  C.  and  pressure  p0. 
If  the  temperature  is  raised  to  t°  C.  at  constant  pressure, 


Now,  if  the  pressure  is  changed  from  p0  to  p,  the  tempera- 
ture being  constant,  the  density  will  be  changed  from  pt  to 
p,  where 


Hence  p  /  p  =  p0  (1  +  j3  t)  /  p0, 

P  _  PO 


01 


Consequently,  the  resulting  density,  p,  at  the  temperature 
t°  C.,  and  the  pressure,  p,  may  be  calculated.  Or,  con- 
versely, if  the  density  is  known  at  any  temperature  and 
pressure,  the  density  at  0°  C.  and  at  any  pressure,  p0,  may 
be  deduced.  In  chemistry  the  volume  of  a  gas  "under 
standard  conditions  "  is  the  volume  which  a  given  mass  of 
the  gas  would  occupy  at  0°  C.,  and  at  a  pressure  equal  to 
76  cm.  of  mercury.  Thus,  substituting  v  /  VQ  =  p0  /  p, 


where  v  is  the  observed  volume  at  pressure  p  and  tempera- 
ture t°  C.  ;  and  VQ  is  the  "  volume  under  standard  condi- 
tions," if  po  is  pressure  of  76  cm.  of  mercury,  p  fp0  is  the 
ratio  of  the  actual  pressure  of  the  gas  measured  in  centi- 
metres of  mercury  to  76. 

Another  deduction  from  the  general  formula  is  that,  if 
the  volume  and  mass  of  a  gas  are  kept  unchanged,  but  the 
temperature  varied,  i.  e.  if  p  =  pQ,  the  pressure  changes 
so  that 

.....     (15) 


214  THEORY  OF  PHYSICS  [CH.  II 

That  is,  the  pressure  at  constant  volume  changes  with  the 
temperature  at  the  same  rate  that  the  volume  does  at  con- 
stant pressure. 

The  value  of  /3  as  found  by  experiment  for  gases  is  nearly 

0.003665  ;  so  that  -~  =  273  nearly.  Substituting  this 
value  in  the  formula 

P  -Po 

p(l 
it  becomes 

P 


0 

But  for  any  one  gas  p0  /  po  is  a  constant  ;  for  p0  is  the  den- 
sity of  the  gas  at  0°  C.  when  the  pressure  is  pQ  ;  and  by 
Boyle's  Law  p0  /  pQ  is  constant  for  a  given  temperature.  Call 

=  R.     It  is  evidently  a  constant  which  can  be  easily 


found  by  experiment  for  any  gas,  and  is  sometimes  called 
the  "  gas-constant  "  for  that  gas.     Then 


P  (273  +  t) 

273  +  t  is  a  number  which  gives  the  reading  which  the 
temperature  t°  on  the  centigrade  scale  would  have,  if 
the  scale  started  from  273°  C.  below  0°  C.  This  number; 
273  +  t,  is  sometimes  written  T,  and  is  called  the  "  absolute  " 
gas-thermometer  temperature.  It  is  evidently  a  constant 
for  all  gases. 

The  formula  then  becomes 


or,  substituting  p  —  m  /  v, 

^  =  Rm  .  ;..     .      (16  a) 


179]  CHANGES   IN  VOLUME  215 

This  is  the  fundamental  law  for  a  gas  ;  and  from  it  all  laws  , 
may  be  derived.     It  is  based  upon  two  experimental  laws, 
Boyle's  and  Charles'. 

[It  is  evident  that  if  T—  0,  either  p  or  v  must  equal  0  ; 
but  this  has  no  physical  meaning.  In  fact,  the  law  applies 
only  to  gases  ;  and  there  is  no  reason  to  believe  that  at 
—  273°  C.  matter  would  exist  in  the  form  of  a  gas.  The 
law  is  not  exactly  true  of  any  actual  gas,  but  expresses  the 
ideal  behavior  of  a  "  perfect  "  gas.] 

179.  Measurement  of  j3.  The  constant  /3  may  be  meas- 
ured in  two  ways  :  the  pressure  may  be  kept  constant  and 
the  volume  measured  at  different  temperatures,  in  which 
case 


or,  the  volume  may  be  kept  constant  and  the  change  in 
pressure  measured  when  the  temperature  changes,  in  which 
case 


With  all  actual  gases  the  values  of  /3  as  found  by  these 
two  methods  differ  slightly,  showing  that  Boyle's  and 
Charles'  laws  are  not  rigidly  true  of  actual  gases. 

Application  of  Principle  of  Stable  Equilibrium.  A  mass  of 
gas  enclosed  in  a  vessel  (e.  g.  a  cylinder  with  movable 
piston)  is  in  stable  equilibrium  if  it  is  in  a  region  of  con- 
stant temperature.  All  gases  expand  if  the  temperature  is 
raised  ;  so  expanding  a  gas  lowers  the  temperature,  com- 
pressing it  raises  the  temperature. 


CHAPTEE   III 

CHANGE   OF  TEMPERATURE 

180.  WHEN  molecular  energy  is  taken  away  from  a 
body  or  added  to  it,  there  is  in  general  what  is  called  a 
change  in  temperature  ;  but  for  equal  amounts  of  energy 
the  temperature  is  affected  differently  for  different  bodies. 

Calorie.  To  investigate  this  difference,  it  would  be  neces- 
sary to  measure  the  change  in  temperature  of  different 
bodies  when  equal  amounts  of  energy  are  given  their  mi- 
nutest portions  so  as  to  produce  heat-effects,  or  to  determine 
the  amount  of  energy  necessary  to  raise  the  temperature 
one  degree.  This  would  be  a  most  difficult  process,  be- 
cause our  ordinary  methods  of  producing  changes  in  tem- 
perature do  not  admit  easily  of  measurement  in  terms  of 
ergs,  the  unit  of  energy.  The  best  method  is  to  take  some 
secondary  unit,  in  terms  of  which  the  energy  going  into 
heat-effects  may  be  easily  measured,  and  whose  value  in 
terms  of  ergs  may  be  once  for  all  determined.  Such  a  unit 
is  the  energy  required  to  raise  the  temperature  of  one  gram 
of  water  from  10°  C.  to  11°  C. ;  and  it  is  called  a  "  calorie  " 
or  a  "thermal  unit  at  10°  C."  Its  value  in  terms  of  ergs 
is  found  by  experiment  to  be  4. 2  X  107 ;  and  it  is  evidently 
a  most  convenient  practical  unit,  in  terms  of  which  to  mea- 
sure the  energy  required  to  produce  in  any  body  rise  in 
temperature  or  any  heat-effect.  The  amount  of  energy 
required  to  raise  the  temperature  of  one  gram  of  water 
one  degree  anywhere  else  than  at  10°  C.  is  not  1  calorie  ; 
but  it  differs  from  that  so  slightly  that  in  ordinary  experi- 
ments the  difference  may  be  neglected. 


181] 


CHANGE   OF  TEMPERATURE 


217 


181.  Specific  Heat.  The  number  of  calories  which  is 
required  to  raise  the  temperature  of  1  gram  of  any  sub- 
stance from  t°  C.  to  (t  +  1)°  C.  is  called  the  "  specific  heat " 
of  that  substance  at  the  temperature  t°.  The  specific  heat 
of  any  substance  is  different  for  different  temperatures ; 
and  so  the  average  specific  heat  is  the  number  of  calories 
required  to  raise  the  temperature  of  1  gram  a  certain  num- 
ber of  degrees  centigrade,  divided  by  the  number  of  degrees. 
The  specific  heat  varies  greatly  for  different  substances. 
Some  few  values  are  given  in  the  following  table  :  — 

TABLE   V 
AVERAGE  SPECIFIC  HEATS 


Alcohol 
Aluminum 

0°-100° 

0.615 
0.2185 

Mercury 
Paraffi  n 

20°-50° 

0.0333 
0.683 

Brass 
Copper 
Glass 
Iron 

0°-100° 
0°-100° 

0.09 
0.0933 
0.20 
0.1130 

Platinum 
Silver 
Tin 
Turpentine 

0°-100° 
0°-100° 
0°-100° 

0.0323 
0.0568 
0.0559 
0.467 

Lead 

0°-100° 

0.0315 

It  is  evident,  of  course,  that  if  c  is  the  specific  heat  of 
any  substance,  the  number  of  calories  required  to  raise  the 
temperature  of  m  grams  1  degree  C.  is  m  c.  And,  if  c  is  the 
average  specific  heat  between  the  temperatures  t°  and  T°, 
the  number  of  calories  required  to  raise  the  temperature 
of  m  grams  that  amount  is  m  c  (T°  — £°).  Conversely,  if 
m  grams  cool  from  T°  to  t°,  that  number  of  calories  must 
leave  the  body,  if  it  returns  to  its  original  condition. 

Some  of  the  energy  which  is  added  to  the  body  when 
the  temperature  is  raised,  is  spent  in  producing  molecular 
changes  which  involve  changes  in  potential  energy  (except 
in  perfect  gases)  and  also  in  doing  external  work ,  so  the 
true  specific  heat  would  be  only  that  portion  of  the  energy 


218  THEORY  OF  PHYSICS  [CH.  Ill 

which  is  spent  in  raising  temperature.  It  is,  however, 
practically  impossible  to  measure  this  quantity  except  in 
the  case  of  gases. 

182.  Specific  Heat  of  Gases.  It  will  be  shown  later  (see 
Art.  186)  that  in  the  more  perfect  gases,  such  as  hydrogen 
and  oxygen,  practically  no  energy  is  required  to  produce  any 
rearrangement  of  their  molecules.  If  the  volume  of  a  gas 
is  then  increased,  the  only  energy  required  is  that  necessary 
to  do  the  external  work,  provided  the  temperature  does  not 
change.  The  number  of  calories  required  to  raise  the  tem- 
perature of  1  gram  of  a  gas  1°  C.,  if  the  volume  is  kept 
constant,  is  called  the  "  specific  heat  at  constant  volume." 
It  is  ordinarily  written  Cv  ;  and  it  is  the  true  specific  heat 
of  the  gas.  The  number  of  calories  required  to  raise  the 
temperature  of  1  gram  of  the  gas  1°  C.  when  the  volume 
increases,  but  the  pressure  is  kept  constant,  is  called  the 
"specific  heat  at  constant  pressure,"  and  is  ordinarily 
written  Cp.  Cp  is,  of  course,  greater  than  Cv  ;  and  the 
difference  Cp  —  Cv,  expressed  in  ergs,  is,  by  what  was  said 
above,  the  amount  of  external  work  done  when  1  gram  of 
the  gas  expands  at  constant  pressure  owing  to  its  temper- 
ature being  raised  1°  C.  This  may  be  easily  calculated. 
By  Formula  16  a,  Chapter  II., 

p  v  =  Rm  T. 

So  that,  if  we  deduce  the  change  in  volume  of  1  gram 
when  the  temperature  is  raised  from  Tto  T  +  1,  the  pres- 
sure being  constant,  we  have 

v  =  R  T 


.  •  .   p  (vi  —  v)  —  R. 

But  the  external  work  done  by  the  gas  is  (by  Art:  104) 
the  product  of  the  pressure  and  the  change  in  volume, 
p  (vi  —  v),  i.  e.  72. 


182] 


CHANGE   OF  TEMPERATURE 


219 


Call  /=4.2  X  107.  Hence  the  number  of  ergs  in 
Op  —  Cv  calories  is  J  (Cp  —  Cv).  Consequently, 

J(C,-CW)=R  ......    (1) 

All  four  of  these  quantities  may  be  measured,  and  the 
formula  is  found  to  be  verified. 

It  may  be  well  to  add  that  for  any  gas  both  Cp  and  Cv 
are  nearly  constant  for  all  temperatures ;  and  so  their  ratio 
is  also.  But  both  C  and  Cv  change  with  the  pressure  ; 
that  is,  if  the  gas  is  compressed,  these  quantities  change. 
For  air  and  carbon  dioxide  (£02)  Cv  increases  if  the  pres- 
sure is  increased ;  while  for  hydrogen  just  the  opposite 
is  the  case.  The  quantity  Cp  /  Cv  is  commonly  called  7  ; 
and  it  may  be  found  by  direct  experiment.  (See  Art. 
119.)  Further,  as  noted  in  Article  109,  the  ratio  of  the 
elasticity  of  a  gas  when  it  is  compressed  very  rapidly  to 
the  elasticity  when  the  gas  is  compressed  slowly  may  be 
proved  by  both  theory  and  experiment  to  be  the  same  as 
the  ratio  Cp  /  Cv. 

Some  values  of  Op,  Cvt  and  7  are  given  in  the  following 
table  ;  and  it  will  be  seen  that  7  as  found  by  experiment 
very  nearly  equals  Op  I  Cv. 

TABLE   VI 
SPECIFIC  HEATS  OF  GASES 


CP 

Cv 

7 

Air          

0.237 

0.171 

1.40+ 

"Argon"         .... 

1.66 

Chlorine            .... 

0.124 

1.33 

C02  

.165 

1.29 

"Helium"       .... 
Hydro  sren                     •     • 

0.340 

.241 

1.74  (?) 
1  41 

Mercury  (vapor)   . 
Nitrogen     

0.244 

1.67 
1  41 

0.217 

1  41— 

220  THEORY  OF   PHYSICS  [CH.  Ill 

183.  Dulong  and  Petit's  Law.     It   lias   been  noticed  by 
several  observers  that  there  is  a  simple  mathematical  connec- 
tion between  the  specific  heats  of  various  substances  and 
their  atomic  weights.     (See  some  text-book  on  Chemistry.) 
One  of  these  relations  was  first  observed  by  Dulong  and 
Petit.     They  found  that  if  the  product  of  the  specific  heat 
of   any   elementary  substance   and   its    atomic  weight  is 
formed,  the  number  is  nearly  the  same  for  all  substances. 
Thus 

Specific  Heat.        Atomic  Weight.        Product. 

Copper 0.0933  63.5  6.1 

Iron  .     .     .     ..;V-..  0.113  54.5  6.2 

Lead 0.0315  207.0  6.4 

Silver 0.0568  108.0  6.2 

This  law  is  equivalent  to  stating  that  the  number  of 
calories  (or  the  amount  of  energy)  required  to  raise  the 
temperature  of  one  atom  of  any  substance  1°  C.  is  the  same 
for  all  substances.  An  exact  agreement  in  all  the  products 
w^as  not  to  be  expected,  because,  as  stated  before,  the 
specific  heat  of  any  substance  is  different  for  different  tem- 
peratures ;  and  so  it  is  impossible  to  choose  those  numer- 
ical values  which  are  exactly  comparable. 

Other  laws  have  been  noticed  for  various  series  of 
compounds,  and  also  for  various  gases. 

184.  Measurement   of   Specific   Heat.    There   are   several 
general  methods  for  the  measurement  of  specific  heats.    The 
best  are  the  following :  — 

a.  Method  of  Mixtures.  The  principle  of  the  method  is 
to  put  into  a  vessel  of  water  the  substance  whose  specific 
heat  is  desired,  and  to  note  the  changes  in  temperature. 
The  initial  temperatures  of  the  water  and  the  substance 
are  different ;  but  their  final  temperature  is,  of  course,  the 
same. 


184]  CHANGE  OF  TEMPERATURE  221 

Let  M  =  mass  of  water, 

mf  =  mass  of  vessel  which  contains  the  water, 
cf  =  specific  heat  of  this  vessel, 
m  =  mass  of  substance  whose  specific  heat  is  desired, 
c  =  the  unknown  specific  heat  of  this  substance, 
T  =  initial  temperature  of  the  water, 
t  =  initial  temperature  of  the  substance  inserted, 
TQ  —  final  temperature. 

A  number  of  calories  equal  to  m  c  (t  —  T0~)  is  taken  away 
from  the  substance  inserted;  a  number  M  (T<>  —  T)  is 
given  the  water,  and  m1  c  (T0  —  T)  to  the  vessel  which 
contains  the  water.  If  no  energy  has  escaped,  the  number 
of  calories  lost  by  the  one  body  must  equal  that  gained  by 
the  other  two.  That  is, 

mc(t-T0)  =  (M  +  mfcf)(T0-  T)  .     .     .     (2) 

The  temperatures  and  the  masses  may  be  easily 
measured ;  so  may  mr  c',  accurately  enough  for  this  experi- 
ment. Consequently  c  may  be  determined. 

m'  c'  is  sometimes  called  the  "  water-equivalent "  of  the 
calorimeter,  or  the  vessel  containing  the  water.  It  may 
be  determined  roughly,  by  pouring  into  the  empty  vessel 
a  known  mass  of  water  at  a  known  temperature,  and 
noticing  the  change  in  temperature. 

Let  M  =  mass  of  water  poured  in, 

T  =  temperature  of  the  water  as  it  is  poured  in, 
t  =  initial  temperature  of  the  vessel, 
TQ  =  final  temperature. 

Then  M  (T  -  T0)  =  m'  cf  (T0  -  t), 

or  m>c'  =  M?^-^ (3) 

J-  0    v 

and  so  it  may  be  measured. 

This  method  for  the  determination  of  the  specific  heat 
is  applicable  to  any  solid  or  liquid  which  is  not  acted  upon 


222  THEORY  OF  PHYSICS  [CH.  Ill 

by  the  water,  and  which  may  be  secured  in  considerable 
amounts.  If  water  does  act  chemically  upon  the  solid  or 
liquid,  some  other  liquid  of  known  specific  heat  may  be 
used. 

In  order  to  avoid  losses  of  energy  by  radiation  or  con- 
duction, the  calorimeter  is  polished  and  carefully  separated 
from  surrounding  bodies  by  some  non-conducting  substance, 
such  as  cork,  air,  or  feathers  ;  and,  further,  the  attempt  is 
made  so  to  choose  the  relative  amounts  of  water  and  the 
substance  inserted  that  (T0+T)/2  shall  equal  the  tem- 
perature of  the  surrounding  air;  that  is,  the  initial  tem- 
perature of  the  water  is  just  as  far  below  the  temperature 
of  the  air  as  the  final  temperature  is  above  it. 

b.  Method  of  fusion  of  Ice.  Experiments  to  be  discussed 
later  (Art.  189)  prove  that,  in  order  to  melt  1  gram  of  ice 
at  0°  C.  into  water  at  0°,  80  calories  are  required.  So  that, 
if  the  substance  whose  specific  heat  is  desired  is  placed 
while  hot  in  a  cavity  of  ice  or  is  surrounded  by  ice,  and 
if  the  amount  of  ice  melted  can  be  measured,  the  specific 
heat  may  be  determined. 

Let  m  =  mass  of  substance  whose  specific  heat  is  desired, 
M  =  mass  of  ice  melted, 
t°  =  initial  temperature  of  substance. 

The  substance  loses  met  calories,  since  it  is  cooled  to  0°  C. ; 
80  M  calories  are  added  to  the  ice.  So,  if  there  is  no  loss 
by  radiation  or  otherwise, 

SOM=mct 

80  M 

and  c  =  -  — (4) 

m  t 

Suitable  forms  of  apparatus  for  the  use  of  this  method 
have  been  devised;  and  it  admits  of  great  accuracy  for 
even  small  amounts  of  liquids  and  solids.  The  best  form 
of  apparatus  is  Bunsen's  ice  calorimeter. 


184]          CHANGE  OF  TEMPERATURE  223 

c.  Method  of  Condensation  of  Steam.    It  is  known,  as  the 
result  of  careful  experiments,  that  536  calories  have  to  be 
taken  away  from  1  gram  of  steam  at  100°  C.,  in  order  to 
make  it  condense  into  water  at  100°  ;    and   the   number 
required  at  other  temperatures  is  also  known.     So,  if  a 
substance  at  a  temperature  below  100°  C  is  placed  in  a 
closed  chest  into  which  steam  at  100°  is  suddenly  admitted, 
there  will  be  a  condensation  of  steam  into  water  on  the  sub- 
stance until  its  temperature  is  raised  to  100°  C. ;  and,  if  the 
condensed  water  can  be  caught  in  a  pan  and  weighed,  ^he 
number  of  calories  taken  away  from  the  steam  can  be  cal- 
culated.    The  substance  is  held  in  some  pan  or  vessel  whose 
mass  and  specific  heat  must  be  taken  into  account. 

Let  m  =  mass  of  substance  whose  specific  heat  is  desired, 

c  =  its  specific  heat, 
mf  =  mass  of  pan  or  vessel, 
c'  =  its  specific  heat, 
M  =  mass  of  steam  condensed, 
t°  =  initial  temperature  of  substance  and  pan. 

Then,  if  the  temperature  is  raised  to  100°  C., 

536  Jf  =  (mc+ mV)  (100 -£)    .     .     .     .     (5) 

m1  c'  may  be  found  by  a  preliminary  experiment  in  which 
there  is  no  substance  on  the  pan;  so  c  itself  may  be 
determined. 

This  method  is  perhaps  the  most  accurate  now  in  use  ; 
and  it  can  obviously  be  applied  to  any  solid,  liquid,  or  gas, 
because  they  can  be  all  enclosed  in  some  hollow  closed  ves- 
sel, like  a  sphere,  which  rests  on  a  pan.  With  a  gas,  this 
method  gives,  of  course,  Cv,  the  specific  heat  at  constant 
volume,  because  the  slight  expansion  of  the  containing 
vessel  can  be  regarded  simply  as  a  correction  to  be  deter- 
mined by  a  preliminary  experiment. 

d.  Method  of  Mow.      In  this  method,  which  is  used  to 
measure  Cpt  the  specific  heat  of   a  gas  at  constant  pres- 


224 


THEORY  OF  PHYSICS 


[CH.  Ill 


sure,  a  gas  is  forced  very  slowly  through  a  long  tube 
which  is  bent  into  two  spirals ;  one  spiral  is  kept  in  a 
vessel,  /,  at  a  constant  temperature ;  the  other  is  in  a 
second  vessel,  II,  containing  water  (or  some  other  liquid). 
The  gas  in  passing  through  the  first  spiral  assumes  the 


FIG.  135. 

temperature  of  vessel  /;  it  then  enters  the  second  spiral, 
and  slowly  raises  the  temperature  of  the  surrounding  water 
in  vessel  //.  When  it  escapes  from  the  spiral,  it  will  have 
the  temperature  of  the  surrounding  water,  if  it  is  drawn 
through  slowly  enough.  Thus,  by  noting  the. temperature 
in  vessel  /,  how  much  gas  has  passed  through,  how  much 
water  is  in  vessel  //,  and  its  changes  in  temperature,  it 
is  possible  to  calculate  Cp  for  the  gas. 


CHAPTEE  IV 

MOLECULAR   CHANGES 

As  stated  in  the  Introduction  and  Chapter  L,  among  the 
most  common  heat-effects  are  those  where  the  potential 
energy  of  the  molecules  is  changed.  Several  illustrations 
will  be  discussed  in  this  chapter. 

EXPANSION 

185.  It  requires  the  addition   of   energy  in  general  to 
make  the  molecules  of   any  body  change   their   relative 
positions  ;   and,  if  after  the  change  the  molecules  return 
to  their  former  positions  and  condition,  the  energy  which 
they  give  out  equals  that  which  they  received  when  the 
first  change  was  produced.     So,  when  any  body  expands, 
some  energy  is,  as  a   rule,  required   to   do   the   internal 
work. 

186.  Internal  Work  in  a  Gas.     This  is  always  true  in  the 
case  of  solids  and  liquids,  but  is  not  so  for  gases ;  or,  at 
least,  the  energy  required  is  very  minute.     This  fact  is 
sometimes  called  Joule's  Law ;   because   Joule,  although 
not   the   first  to  prove    the   statement,  yet  was  the   first 
to  show  its   great   importance  and  to  test  it   most   care- 
fully.    Joule  placed  in  a  bath  of  water  two  hollow  metal 
vessels,  connected  by  a  tube  with  a  closed  stop-cock  ;  one 
vessel  contained  a  gas  at  a  high  pressure,  the  other  was 
comparatively  empty.    He  observed  that,  when  the  cock 
was  opened,  thus  allowing  the  gas  to  expand  from  one 
vessel  into  the  other,  there  was  no  change  in  the  average 
temperature  of  the  water-bath.     This  proved  that  no  heat- 

8 


226 


THEORY  OF  PHYSICS 


[CH.  rv 


FIG.  136. 


energy  left  the  gas  or  was  added  to  it.  In  other  words,  no 
energy  was  required  to  make  the  gas  expand.  It  will  be 
noticed  that  the  gas  did  no  external  work  ;  the  only  change 

was  one  of  volume.  Of 
course,  if,  as  a  gas  expands, 
it  does  external  work,  its 
temperature  will  fall.  And 
in  Joule's  experiment  the 
temperature  of  the  gas 
which  remains  in  the  vessel 
where  the  gas  was  under 
pressure  is  found  to  be 
lower  after  the  expansion, 
because  it  has  done  work 
in  driving  the  rest  of  the  gas  out  into  the  other  vessel, 
where  the  temperature  is  found  to  be  increased.  But 
the  loss  of  energy  of  the  one  portion  exactly  equals  the 
gain  of  the  other,  and  the  average  temperature  of  the 
•entire  gas  is  not  changed,  proving,  as  just  stated,  that 
changes  in  the  intrinsic  energy  of  a  gas  are  manifest  as 
changes  in  temperature. 

More  accurate  experiments  show  that  in  all  gases  mi- 
nute changes  in  temperature  are  produced  when  there  is 
-expansion ;  but  these  need  not  be  taken  into  account  in 
any  ordinary  calculation.  Hydrogen  is  slightly  heated  on 
expansion  ;  all  other  gases  are  cooled. 

The  above  statement  as  to  the  nature  of  a  gas  is  equiva- 
lent to  saying  that  its  intrinsic  energy  (see  Art.  167)  de- 
pends on  the  temperature  alone,  not  on  its  volume  or 
pressure. 

FUSION 

1ST.  Fusion  is  the  name  given  to  the  process  by  which 
a  solid  passes  into  the  liquid  condition ;  the  reverse  pro- 
cess is  called  solidification.  These  processes  obviously 
involve  molecular  changes ;  and,  just  as  energy  must  be 


187]  MOLECULAR  CHANGES  227 

added  to  produce  fusion,  so  the  molecules  lose  energy  in 
solidification. 

Fusion-Point.  Most  substances,  under  standard  condi- 
tions, will  begin  to  fuse  or  to  solidify  at  a  definite  tem- 
perature. This  is  true  of  all  crystals.  If  heat-energy  is 
applied  to  such  a  solid,  its  temperature  will  continue  to 
rise  until  a  certain  definite  temperature  is  reached,  and 
then  further  addition  of  heat-energy  produces  no  change  in 
temperature,  but  the  solid  melts.  This  definite  tempera- 
ture is  known  as  the  "  fusion-point "  of  that  substance  ;  and, 
as  long  as  the  process  of  fusion  continues,  the  temperature 
does  not  change.  If  the  liquid  is  cooled  until  it  begins  to 
solidify  in  crystals,  the  temperature  at  which  this  process 
begins  is  the  same  as  that  at  which  the  crystal  fused,  viz. 
the  fusion-point ;  and,  until  the  process  ends  and  all  the 
liquid  is  solidified,  the  temperature  does  not  change.  (Of 
course,  this  temperature  is  that  of  the  entire  mixture  of 
liquid  and  solid,  which  must  be  well  stirred.)  At  this 
temperature,  then,  of  the  fusion-point,  the  solid  and  its 
liquid  are  in  equilibrium  together.  As  long  as  the  pressure 
does  not  change,  the  fusion-point  remains  the  same.  So  it 
may  be  said  that  the  solid  and  liquid  are  in  equilibrium 
together  at  a  definite  temperature  which  depends  only  upon 
the  pressure. 

To  start  the  solidification  of  a  liquid,  a  nucleus  of  some 
kind  is  often  necessary,  or  a  crystal  of  the  same  form  as 
that  of  the  solid  which  is  to  be  formed.  Shaking  will 
sometimes  hasten  the  process  by  helping  the  molecules  to 
rearrange  themselves. 

Some  solids,  especially  the  waxes,  do  not  remain  at  the 
same  temperature  during  the  process  of  fusion,  and  the 
temperature  at  which  fusion  begins  is  not  that  at  which 
solidification  does. 

Determination  of  Fusion-Point.  Various  methods  for  the 
determination  of  the  fusion-point  of  any  solid  are  given  in 
laboratory  manuals ;  but  the  principle  made  use  of  in  all 


228  THEORY  OF  PHYSICS  [CH.  IV 

is  to  determine  that  temperature  at  which  the  solid  and 
the  liquid  are  in  equilibrium  together.  In  the  case  of 
those  solids  like  paraffin,  whose  fusion-  and  solidification- 
temperatures  are  different,  the  average  of  these  two  tem- 
peratures is  often  taken  as  the  fusion-point  or  "  melting- 
point." 

Application  of  Principle  of  Stable  Equilibrium.  If  a  solid 
and  its  liquid  are  in  contact  at  their  fusion- point,  e.  g.  ice 
and  water  at  0°  C.,  they  are  in  stable  equilibrium.  In  all 
known  cases  solidification  results  in  the  production  of  heat- 
energy  ;  therefore,  if  heat-energy  is  directly  applied,  there 
will  be  fusion,  the  opposite  of  solidification. 

188.  Change  in  Volume.  When  any  solid  fuses,  there  is 
always  a  change  in  volume.  Some  substances  expand  on 
fusion,  e.  g.  wax,  bismuth ;  while  others  contract,  e.  g.  ice, 
cast-iron,  brass.  If  a  solid  floats  on  its  liquid,  it  contracts 
on  fusion,  because  the  density  in  the  solid  state  is  less  than 
in  the  liquid  state.  While,  if  a  solid  sinks  in  its  liquid, 
the  opposite  is  true.  It  is  obvious  that  sharp  castings 
can  be  taken  with  only  those  substances  which  expand 
on  solidification. 

Effect  of  Pressure  on  Fusion.  It  was  stated  in  Article  187 
that  under  standard  conditions  the  fusion-point  was  a 
definite  temperature  for  any  one  solid.  If  the  pressure 
changes,  though,  the  fusion-point  is  not  the  same.  The 
pressure  which  is  required  to  produce  even  the  least  measur- 
able change  in  the  fusion-point  is,  however,  enormously 
great  for  ordinary  solids ;  and  so  the  effects  of  ordinary 
atmospheric  changes  in  pressure  are  too  minute  ever  to  be 
noticed.  An  increase  in  pressure  of  one  atmosphere  pro- 
duces a  change  of  O.°0075  C.  in  the  fusion-point  of  ice. 

At  great  pressures,  however,  changes  are  produced ;  and 
there  is,  naturally,  a  connection  between  the  change  in 
volume  on  fusion  and  the  effect  of  the  increased  pressure. 
If  a  solid  expands  on  fusion,  an  increase  in  pressure  would 
tend  to  prevent  the  change ;  and  so  the  temperature  at 


189]  MOLECULAR  CHANGES  229 

which  fusion  begins  is  higher  than  it  is  normally,  i.  e.  the 
fusion-point  is  raised.  If,  on  the  other  hand,  the  solid  is 
one  which  contracts  on  fusion,  the  increase  in  pressure 
helps  the  process,  and  the  temperature  to  which  it  is  neces- 
sary to  raise  the  solid  before  it  melts  is  lower  than  it  is 
normally,  i.  e.  the  fusion-point  is  lowered.  Therefore,  since 
ice  contracts  on  fusion,  its  melting-point  is  lowered  by  an  in- 
crease in  pressure.  So,  if  two  pieces  of  ice,  whose  surfaces 
are  at  0°  under  ordinary  conditions,  are  forced  together 
with  a  great  pressure  (e.  g.  at  two  points),  the  melting- 
point  is  lowered,  the  ice  melts  at  the  point  where  the  pres- 
sure is,  and  the  water  thus  formed  yields  to  the  pressure* 
and  flows  away.  But  its  temperature  is  lower  than  0°C.> 
because  to  produce  fusion  heat-energy  is  taken  from  neigh- 
boring points  and  added  to  the  ice,  and  so  it  almost  instantly 
freezes  again.  This  process  is  known  as  "  regelation ; " 
and  it  explains  at  once  the  motion  of  glaciers  over  steep 
and  rocky  tracks,  the  formation  of  snow-balls  under  the 
pressure  of  the  hands,  etc. 

189.  "  Latent-Heat  of  Fusion."  As  stated  above,  energy 
must  be  added  to  a  solid  in  order  to  make  it  fuse,  and  the 
number  of  calories  required  to  make  1  gram  of  any  sub- 
stance fuse  at  a  fixed  temperature  is  called  the  "  latent-heat 
of  fusion  "  of  that  substance  at  that  temperature.  This 
energy  does  work  in  producing  molecular  rearrangements  ; 
and,  if  a  liquid  solidifies,  it  emits  an  equal  amount  of 
energy.  (A  tub  of  water  is  sometimes  placed  in  a  conser- 
vatory on  a  night  when  a  frost  is  expected,  because,  as  the 
water  freezes,  it  gives  up  heat-energy  to  the  surrounding 
room.) 

The  latent-heat  of  fusion  of  any  substance  is  generally 
determined  by  using  the  method  of  mixtures,  described  for 
measuring  specific-heats,  although  the  method  of  condensa- 
tion of  steam  may  be  used. 

The  following  table  gives  the  fusion-points  and  the  latent- 
heats  of  fusion  of  a  few  substances  :  — 


230 


THEORY  OF  PHYSICS 
TABLE   VII 


[CH.  IV 


Fusion-Point. 

Latent-Heat 
of  Fusion. 

Copper   

1050°  C 

Ice                     „     .     .     .    • 

0° 

80 

Iron   .     .     .     .*.    .     .'    . 
Lead       .                 * 

1400°-1600° 

325° 

23-33 
5  86 

Mercury      ..... 
Sulphur       ... 

OQO 
—  Off 

115° 

2.82 
9  37 

Zinc        

415° 

28. 

190.  Effect  of  Dissolved  Substances.  When  there  is  any 
substance  dissolved  in  the  liquid,  the  temperature  of  solid- 
ification is  changed ;  a  lower  temperature  is  required  than 
would  be  for  the  pure  liquid.  In  general,  when  the  liquid 
solidifies,  it  does  so  leaving  behind  it  the  impurity.  But 
in  certain  cases,  e.  g.  common  salt,  NaCl,  in  water,  for  a 
definite  percentage  of  salt  in  solution,  there  is  a  definite 
temperature  at  which  this  solution  solidifies  as  a  solid 
of  salt  and  ice  together.  Such  solutions  are-  called  "  cryo- 
hydrates." 

When  common  salt  is  put  on  ice  or  snow,  there  is  fusion 
generally,  because  the  surrounding  temperature  is  not  low 
enough  to  keep  the  mixture  of  ice  and  salt  in  a  solid 
form ;  so  the  ice  melts  and  the  salt  is  dissolved  until  the 
temperature  is  lowered  to  that  of  equilibrium  of  salt  and 
ice  in  the  liquid  and  solid  conditions.  In  each  of  these 
processes,  the  melting  of  the  ice  and  the  solution  of  the 
salt,  energy  is  taken  away  from  surrounding  bodies ;  and 
so  their  temperatures  are  lowered.  Mixtures  of  ice  and 
common  salt  are  therefore  called  "  freezing  mixtures." 

There  is  an  intimate  connection  between  the  nature  and 
quantity  of  the  substance  dissolved  in  the  liquid,  the 
liquid  itself,  and  the  lowering  of  the  fusion-point ;  but  the 
laws  are  too  complicated  to  be  given  here. 


192]  MOLECULAR  CHANGES  231 

EVAPORATION 

191.  Evaporation  is  the  name  given  to  the  process  by 
which  a  liquid  passes  into  the  gaseous  condition.     When 
liquids    evaporate,    they    are    generally     said     to    form 
"vapors,"   although   it   is    quite   impossible   to   give   any 
definition  which  will  distinguish  a  vapor  from  a  gas. 

Vapors.  Unlike  the  process  of  fusion,  which  does  not  begin 
until  the  temperature  of  the  solid  is  raised  to  a  definite  de- 
gree, the  process  of  evaporation  is  going  on  all  the  time 
from  the  surface  of  a  liquid,  no  matter  what  its  tempera- 
ture is.  If  a  vessel  of  any  liquid  is  placed  in  a  closed  space 
(e.  g.  under  a  bell-jar),  it  is  observed  that  the  liquid  evapo- 
rates gradually,  but  that  finally  the  process  stops ;  that  is, 
the  mass  of  the  liquid  remains  constant.  When  this  limit 
is  reached,  the  liquid  and  its  vapor  are  in  equilibrium ; 
and  the  vapor  is  said  to  be  "  saturated."  Before  this  final 
stage  is  attained,  the  vapor  is  called  "  unsaturated." 

192.  Laws  of  Saturated  Vapor.     Unsaturated  vapors  are 
simply  imperfect  gases ;   and   they    obey    approximately 

f)  V 

the  laws  of  Boyle  and  Charles,  ±-=  —  E  m.     But  saturated 

vapors  are  quite  different,  because  the  moment  such  a 
vapor  is  compressed  some  vapor  condenses ;  and,  if  it  is 
expanded,  it  ceases  to  be  saturated,  unless  it  is  over  a 
vessel  of  the  liquid,  in  which  case  more  liquid  will  evapo- 
rate so  as  to  make  the  vapor  saturated.  This  saturated 
vapor  exerts  a  certain  pressure ;  and  it  is  proved  by 
experiment  that  the  process  of  evaporation  comes  to  a 
state  of  equilibrium  at  a  definite  pressure  for  any  given 
temperature.  In  other  words,  if  the  pressure  is  constant, 
there  is  a  definite  temperature  at  which  a  liquid  and  its 
vapor  will  be  in  equilibrium ;  and,  conversely,  for  a  given 
temperature  there  is  a  definite  pressure  of  equilibrium. 
Stated  in  a  different  way,  the  pressure  of  saturated  vapor 
depends  only  upon  the  temperature  ;  and  conversely. 


232 


THEORY  OF  PHYSICS 


[CH.  IV 


Equilibrium  over  Curved  Surfaces.  It  should  be  stated 
that  for  a  given  temperature  the  pressure  of  saturated 
vapor  depends  also  upon  the  shape  of  the  surface  of 
the  liquid  with  which  it  is  in  contact.  There  is  a  defi- 
nite pressure  for  a  plane  surface ;  the  pressure  for  a  con- 
cave surface  is  less  than  this ;  that  for  a  convex  surface, 
greater. 

If  a  liquid  is  placed  in  a  closed  space,  and  if  a  capillary 
tube  is  placed  in  the  liquid,  the  liquid  will  either  rise  or 
fall,  and  there  will  finally  be  equilib- 
rium  between  the  vapor  and  the  sur- 
faces  of  the  liquid.  If  the  liquid  rises, 
i.  e.  if  the  surface  in  the  tube  is  concave, 
the  pressure  of  the  vapor  on  it  is  less 
than  that  on  the  plane  surface  of  the  rest 
of  the  liquid  by  an  amount  p  g  h,  where 
p  is  the  density  of  the  vapor.  If  the 
liquid  falls,  i.  e.  if  the  surface  is  convex, 
the  pressure  of  the  vapor  is  greater 
than  that  on  the  plane  surface  by  this 
amount.  But  h,  the  rise  or  fall,  varies 
inversely  as  the  radius  of  the  curved  sur- 
face (see  Art.  97).  And  so  these  variations  in  vapor- 
pressure  are  very  slight  unless  the  radius  of  curvature  is 
extremely  minute. 

This  law  may  be  stated  in  another  way :  the  pressure 
which  is  required  to  hold  a  drop  of  a  liquid  in  equilibrium, 
to  keep  it  from  evaporating,  is  greater  than  that  required 
to  keep  a  plane  surface  in  equilibrium.  Further,  the  less 
the  radius,  the  greater  is  the  pressure  required.  So  that 
in  a  cloud  of  drops  of  water  of  different  sizes,  the  smaller 
drops  will  evaporate  and  disappear  unless  the  pressure  is 
very  great ;  and  the  larger  drops  will  grow  in  size. 

There  are  two  experimental  methods  of  studying  the 
laws  of  equilibrium  of  saturated  vapor  and  its  liquid, — 
the  "statical"  and  the  " dynamical." 


FIG.  137. 


193] 


MOLECULAR  CHANGES 


233 


193.  The  Statical  Method.  Some  liquid  is  carefully  forced 
into  a  mercury  barometer  which  dips  into  a  deep  bath  of 
mercury.  The  liquid  rises  to  the  top  of  the  mercury 
column ;  and  some  evaporates,  producing  a  certain  pressure, 
and  so  forcing  the  mercury  down.  The  distance 
the  mercury  falls  marks  the  pressure  of  the  va- 
por. Or,  if  H  is  the  height  of  the  barometer  be- 
fore the  vapor  is  inserted,  and  h  the  height  of 
the  column  of  mercury  over  which  the  vapor  is, 
the  pressure  of  the  vapor  is  pg  (H — h),  where 
p  is  the  density  of  mercury.  (Corrections  must 
be  applied  for  the  effects  produced  by  the  weight 
of  the  liquid  and  the  variations  in  the  tempera- 
ture of  the  mercury.)  The  temperature  of  the 
vapor  may  be  kept  constant  by  surrounding 
the  upper  end  of  the  closed  tube  with  a  bath ; 
and  the  temperature  of  this  bath  may  be  changed 
from  time  to  time  so  as  to  study  the  effect  of 
different  temperatures. 

Two  facts  can  now  be  observed.     If  the  tem- 
perature is  kept  constant,  the  pressure  of  the 
vapor  remains  the  same,  however  the  tube  is 
raised  or  lowered  in  the  mercury  bath  so  as  to 
vary  the  volume.     Thus,  if  the  volume  is  in- 
creased, more  liquid  evaporates ;   while,  if   the    FlG  13g 
volume  is  decreased,  some  vapor  condenses.    Also, 
if  the  temperature  is  changed,  the  vapor-pressure  varies. 
In  the  case  of  all  liquids  at  ordinary  temperatures,  an  in- 
crease in  temperature  requires  an  increased  pressure  in 
order  to  keep  the  vapor  and  liquid  in  equilibrium,  and  a 
decrease   in   temperature   requires   a   decreased   pressure. 
Thus,  if  the  temperature  is  raised,  more  liquid  evaporates ; 
if  the  temperature  falls,  some  vapor  condenses. 

So,  evaporation  can  be  produced  either  by  raising  the 
temperature  of  the  liquid  and  vapor,  or  by  increasing  the 
volume  open  to  the  vapor.  Similarly,  condensation  can  be 


234 


THEORY   OF  PHYSICS 


[CH.  IV 


produced  by  lowering  the  temperature  or  by  decreasing 
the  space  in  which  the  vapor  is.  Illustrations  of  this 
principle  are  furnished  in  the  deposition  of  dew,  the  con- 
densation of  steam  in  the  "  condenser "  of  a  steam-engine, 
the  condensation  of  vapors  in  general  in  various  forms  of 
condensers  such  as  Liebig's.  Condensation,  of  course,  is 
assisted  by  the  presence  of  points  or  nuclei  around  which 
the  drops  of  liquid  may  form,  as  was  explained  in  the  sec- 
tion on  Capillarity  (see  Art.  97). 

194.   The  Dynamical  Method.     If  the   temperature  of   a 
liquid    is    raised    enough,    bubbles    of    vapor   are  formed 

throughout  the  liquid,  es- 
pecially at  the  surface  of 
the  vessel  where  it  is  be- 
ing heated.  This  process 
is  called  "  ebullition,"  or 
"  boiling."  It  is  assisted 
by  the  presence  in  the 
liquid  of  nuclei  or  parti- 
cles of  dissolved  gases,  or 
by  any  roughnesses  on  the 
surface  of  the  vessel,  be- 
cause this  helps  the  forma- 
tion of  the  capillary  sur- 
faces of  the  bubbles.  The 
vapor  in  these  bubbles  is 
obviously  saturated,  and 
the  pressure  of  this  vapor 
must  be  equal  to  (or  slightly 
greater  than)  the  pressure 
on  the  surface  of  the  liquid,  if  the  boiling  is  going  on  freely. 
A  thermometer  may  be  hung  above  the  liquid  and  im- 
mersed in  the  vapor  so  as  to  measure  its  temperature,  as 
shown  in  the  figure.  The  pressure  on  the  surface  of  the 
liquid  may  be  measured,  and  it  may  be  varied  at  will  by 
exhausting  or  compressing  the  air  above  the  liquid.  It  is 


FIG.  139. 


LOWER  DIVISION 


194] 


MOLECULAR  CHANGES 


235 


found  on  observation  that  for  a  definite  pressure  there  is 
a  definite  temperature  at  which  boiling  begins,  and  which 
the  vapor  maintains  as  long  as  the  process  continues. 
This  temperature  is  called  the  "  boiling-point."  In  other 
words,  the  temperature  of  saturated  vapor,  i.  e.  of  equi- 
librium between  the  liquid  and  its  vapor,  is  a  constant 
for  a  given  pressure.  If  the  pressure  is  increased,  so  is 
the  boiling-point,  and  vice  versa. 


1500 


1CCO 


WATER-VAPOR  PRESSURE  CURVE 
FIG.  140. 

The  pressure  which  corresponds  to  a  given  temperature 
is  found  to  be  the  same,  whether  determined  by  the  statical 
method  or  the  dynamical.  The  connection  between  corre- 
sponding temperature  and  pressure  may  be  shown  best  by 
a  curve,  called  a  "  vapor-pressure  curve,"  where  distances 


236 


THEORY  OF  PHYSICS 


[CH.  IV 


in  a  horizontal  direction  give  the  temperature  in  centigrade 
degrees,  and  vertical  distances  give  the  pressure  in  centi- 
metres of  mercury.  The  curve  as  shown  is  for  water- 
vapor  or  steam. 

Water  boils  at  100°  C.  when  the  pressure  is  76  cm.  of 
mercury  ;  and,  for  slight  variations  of  pressure,  it  is  found, 
as  noted  in  Article  168,  that  a  difference  of  pressure  of 
1  cm.  produces  a  difference  in  the  boiling-point  of  0°.366  C. 

The  following  table  gives  the  vapor-pressures  of  steam 
at  different  temperatures  :  — 

TABLE   VIII 
VAPOR-PRESSURE  OF  WATER- VAPOR 


Temperature, 
Centigrade. 

Pressure  in  centimetres 
of  Mercury. 

Temperature, 
Centigrade. 

Pressure  in  centimetres 
of  Mercury. 

-5° 

0.316 

98° 

70.713 

0°       • 

0.457 

99° 

73.316 

5° 

'0.651 

99°.2 

73.846 

10° 

0.914 

99°.  4 

74.380 

20° 

1.736 

99°.6 

74.917 

30° 

3.151 

99°.  8 

75.457 

40° 

5.486 

100°. 

76.000 

50° 

9.198 

100°.2 

76.547 

60° 

14.888 

100°.4 

77.100 

70° 

23.331 

100°.6 

77.650 

80° 

35.487 

100°.8 

78.207 

90° 

52.547 

101° 

78.767 

95° 

63.600 

102° 

81.609 

97° 

68.188 

110° 

107.54 

195.  Latent  Heat  of  Evaporation.  Energy  is  of  course 
required  to  produce  evaporation ;  and,  conversely,  if  con- 
densation takes  place,  energy  is  given  out.  The  fact  that 
energy  is  taken  from  surrounding  objects  when  a  liquid 


195] 


MOLECULAR  CHANGES 


237 


evaporates  is  familiar  to  every  one  by  the  cold  sensation 
felt  when  water  evaporates  from  one's  hand  or  body. 
Water  may  be  frozen  as  a  consequence  of  rapid  evapora- 
tion ;  and  extremely  low  temperatures  may  be  secured  by 
producing  evaporation  from  various  liquids,  which  them- 
selves boil  at  low  temperatures. 

The  number  of  calories  required  to  evaporate  1  gram  of 
any  liquid  at  a  given  temperature  is  called  the  "  latent  heat 
of  evaporation  "  of  that  liquid  at  that  temperature.  It  is 
usually  determined  by  a  modification  of  the  method  of 
mixtures  (see  Art.  184).  A  known  number  of  grams  of 
the  vapor  are  conducted  at  a  known  temperature  into  a 
known  number  of  grams  of  the  liquid,  and  the  rise  in  tem- 
perature of  the  liquid  is  noted. 

The  following  table  contains  the  latent  heats  of  evapora- 
tion of  some  liquids  and  their  boiling-points,  at  standard 
atmospheric  pressure :  — 

TABLE   IX 


Boiling-Point. 

Latent  Heat. 

Alcohol  (ethyl)    .     .     . 
Carbon  Dioxide   . 
Chloroform      .... 
Cyanofiren        .... 

78°.4  C. 
-80° 
61°.20 

—20°.  7 

209 

72  at  -25° 
58.5 
103  at  0° 

Ether  (ethyl)        .      .     . 
Hydrogen                        . 

34°  .9 
—243° 

90 

IVTercury                . 

357° 

62 

Oxygen      
Water         

-184° 
100° 

535.9 

Application  of  Principle  of  Stable  Equilibrium.  A  liquid 
and  its  saturated  vapor  are  in  stable  equilibrium  at  a 
definite  temperature  and  pressure.  Condensation  of  vapor 
always  develops  heat-energy  ;  consequently,  if  heat-energy 


238  THEORY  OF   PHYSICS  [CH.  IV 

is  added,  the  opposite  of  condensation  must  be  produced, 
that  is,  evaporation. 

196.  Barton's  Law  for  Mixtures  of  Vapors.     If  two  vapors 
or  gases  are  enclosed  in  the  same  space,  it  has  been  found 
by  experiment  that  their  mixture  is  uniform,  and  that  each 
vapor  or  gas  produces  (nearly)  the  same  pressure  as  it  would 
by  itself  if  the  other  were  not  there,  provided  that  the 
two  gases  or  vapors  do  not  act  on  each  other  chemically. 
This  law  is  sometimes  called  "  Dalton's  Law  of  Mixtures." 

197.  Atmospheric  Moisture.     There  is  always  some  water- 
vapor  in  the  atmosphere ;  and,  if  the  temperature  is  low- 
ered enough,  the  vapor  becomes  saturated,  and  then  for  a 
slightly  lower  temperature  dew  is  formed.     This  tempera- 
ture is  called  the  "  Dew-Point.'v    The  "  hygrometric  state  " 
at  any  temperature  of  the  air  is  the  ratio  of  the  actual 
amount  of  moisture  in  the  air  to  the  amount  that  the  air 
could   possibly   hold   at   the    existing   temperature.      Its 
value  may  be  easily  found  by  experiment,  but  the  method 
is  not  important  enough  for  discussion  here. 

198.  Spheroidal  State.     If  a  small  amount  of  liquid  is 
dropped  on  a  very  hot  metal  plate,  it  may  be  observed  that 
the  liquid  does  not  touch  the  plate ;  it  is  supported  on  a 
cushion  of  its  own  vapor  which  is  produced  by  the  hot 
plate.      This  condition  is   called   the   "spheroidal   state" 
from  the  shape  which  the  drop  of  liquid  assumes.     The 
plate  must  of  course  be  extremely  hot,  e.  g.  red-hot  silver 
or  platinum. 

For  a  similar  reason  a  hand  moistened  with  water  may 
be  dipped  with  impunity  into  a  pot  of  melted  lead,  if  the 
temperature  of  the  lead  is  very  hot.  . 

199.  Effect  of  Dissolved  Substances.     If  any  substance  is 
dissolved  in  a  liquid,  the  boiling-point  of  the  solution  is 
raised ;  that  is,  the  temperature  of  equilibrium  correspond- 
ing to  any  pressure  is  increased.     Consequently  the  pres- 
sure which  corresponds  to  any  temperature  is  lowered.   The 
vapor  formed  over  the  solution  is,  as  a  rule,  the  vapor  of 


200] 


UN/ 

MOLl 


IftXi 

'ARTMENT  OF  PHYSICS 
the  pure  liquid.  Many  interesting  facts  and  laws  have 
been  observed  as  to  the  nature  of  solutions  and  their  boil- 
ing-points and  vapor-pressures ;  but  they  cannot  be  dis- 
cussed here. 

200.  Liquefaction  of  Gases.  All  known  gases  with  one 
or  two  exceptions,  have  been  liquefied,  and  there  is  no  rea- 
son why  these  gases  may  not  be  liquefied  by  suitable  means. 
The  method  used  is  to  lower  the  temperature  of  the  gas  as 
far  as  possible  and  then  to  compress  it.  It  has  been  shown 
that,  unless  the  temperature  is  low  enough,  no  amount  of 
compression  will  ever  liquefy  the  gas.  There  is,  in  fact, 
for  each  gas  a  definite  temperature,  known  as  the  "  critical 
temperature,"  below  which  the  gas  must  be  before  it  can 
be  liquefied.  Some  critical  temperatures  are  given  in  the 
following  table :  — 


TABLE   X 
CRITICAL  TEMPERATURES 


Alcohol     .     . 

243°  .  6  C. 

Hydrogen 

-234° 

Ammonia  . 

130° 

Nitrogen  .     . 

-146° 

"  Argon  "      , 

-121° 

Oxygen     .     . 

-119° 

C02      ... 

31° 

S02      ... 

156° 

Chloroform    . 

260° 

Water  .     .     . 

365° 

Isothermals.  The  entire  process  of  liquefaction  may  best 
be  studied  by  a  consideration  of  certain  curves  which  can 
be  drawn  to  represent  the  behavior  of  liquids  and 
vapors. 

An  "  isothermal  "  is  a  curve  which  is  the  locus  of  points 
representing  the  condition  of  a  given  body  while  its  tem- 
perature is  kept  constant,  but  its  other  properties  varied. 
Usually  the  varying  properties  considered  are  the  pressure 
and  volume ;  and  the  curves  are  drawn  so  that  distances  in 
a  horizontal  direction  represent  volumes  ;  and  in  a  vertical 
one  pressures.  The  isothermal  for  a  perfect  gas  is  as  shown ; 


240 


THEORY  OF   PHYSICS 


[CH.  IV 


because,  for  a  gas  at  constant  temperature,  p  v  =  const. 
And  the  isothermal  for  a  saturated  vapor  is  a  horizontal 
line ;  because  at  constant  temperature  the  pressure  is  con- 
stant, being  independent  of  the  volume.  For  a  liquid,  the 
isothermal  is  a  line  nearly  vertical,  because,  in  order  to 
produce  even  a  slight  decrease  in  volume,  an  enormous 
increase  in  pressure  is  necessary. 


p 

SATURATED  VAPOR                           LIQUID 
FIG.  141. 

F 


PERFECT  GAS 


When  a  gas  is  at  a  temperature  lower  than  the  critical 
one,  the  process  of  complete  liquefaction  is  in  three  stages : 
the  gas  is  compressed  at  constant  temperature  until  it  be- 
comes saturated,  as  it 
will  at  a  definite  pres- 
sure, depending  upon 
the  temperature ;  fur- 
ther decrease  in  vol- 
ume produces  con- 
densation of  the  vapor 
in  liquid  form,  but  no 
change  in  pressure, 
since  the  temperature 
is  constant ;  this  con- 
densation will  con- 
tinue until  for  a  defi- 
nite volume  all  the  vapor  has  been  liquefied  ;  any  further 
decrease  in  volume  requires  now  a  great  increase  in  pres- 
sure. So  the  isothermal  for  the  entire  process  is  a  combi- 
nation of  the  three  curves  described  above. 


FIG.  142. 


200] 


MOLECULAR  CHANGES 


241 


P 
100-1- 


ATM. 


ISOTHERMALS  OF  CARBON  DIOXIDE 
FIG.  143. 


For  a  different  temperature  there  would  be  a  different 
isothermal ;  and  the  accompanying  figure  shows  a  series  of 
isothermals  as  actually  observed  for  carbonic  acid  gas, 
C0a.  The  critical  temperature  is  that  which  corresponds 


242  THEORY  OF  PHYSICS  [CH.  IV 

to  the  first  isothermal  which  has  no  horizontal  portion ; 
because  it  is  only  when  there  is  a  horizontal  portion  that 
any  liquefaction  may  be  seen.  The  locus  of  the  points 
which  mark  the  right-hand  ends  of  the  horizontal  lines, 
i.  e.  the  points  of  saturated  vapor,  is  called  the  "  vapor 
curve."  The  locus  of  the  left-hand  ends,  i.  e.  the  points  of 
complete  condensation,  is  called  the  "  liquid  curve."  These 
two  curves  meet  in  a  point  on  the  critical  isothermal  known 
as  the  "  critical  point."  The  pressure  corresponding  to  this 
point  is  called  the  "  critical  pressure." 

A  liquid  in  a  condition  represented  by  a  point  on  the 
diagram  near  the  critical  point  is  said  to  be  in  a  "  critical 
state,"  because  any  slight  change  may  produce  great  dis- 
turbances, e.  g.  if  the  temperature  is  raised,  the  volume 
being  kept  constant,  all  the  liquid  instantly  becomes 
vaporized. 

SUBLIMATION 

201.  Sublimation   is  a   name  given  to  the   process  by 
which  a  solid  becomes  a  vapor  without  passing  through 
the  intermediate  liquid  form.    Many  solids  do  this.    Snow, 
camphor,  sulphur,  and  others  often  evaporate  directly. 

Certain  laws  have  been  observed.  The  most  important 
is  that,  corresponding  to  any  definite  temperature,  there  is 
a  certain  pressure  at  which  the  solid  and  its  vapor  are  in 
equilibrium,  and  conversely.  The  pressure  which  thus 
corresponds  to  a  definite  temperature  may  be  measured  by 
either  the  statical  or  the  dynamical  method.  The  latent 
heat  of  sublimation  has  been  measured  also. 

DISSOCIATION 

202.  Dissociation  is  the  name  given  to  that  process  by 
which  the  molecules  of  a  substance  are  decomposed  into 
simpler  parts.     It  may  be  produced  in  various  ways ;  but 
of  course  there  are  energy-changes,  because  the  potential 
energy  of  a  molecule  is  not  the  same  as  that  of  its  disso- 
ciated parts. 


203]  MOLECULAR   CHANGES  243 

One  method  of  producing  dissociation  of  a  gas  is  to  raise 
its  temperature.  Under  these  conditions  many  gases  are 
partially  dissociated.  If  the  pressure  of  the  mixture  of 
gases,  the  original  one  and  its  parts,  is  observed,  it  is  found 
that  corresponding  to  any  temperature  there  is  a  definite 
pressure  at  which  the  dissociation  stops,  i.  e.  a  definite 
pressure  of  equilibrium ;  and  the  number  of  calories  re- 
quired to  dissociate  1  gram  at  a  given  temperature  is  called 
the  "  heat  of  dissociation  "  at  that  temperature. 

The  amount  of  energy  required  to  dissociate  a  compound, 
or  the  amount  set  free  when  the  compound  is  formed  (if 
such  is  the  case),  may  be  measured  by  ordinary  calorimet- 
ric  methods.  Some  compounds  absorb  energy  when  they 
are  formed.  The  number  of  calories  produced  or  required, 
if  1  gram  of  the  compound  is  formed,  is  called  the  "  heat  of 
combination  "  of  the  substance. 

There  is  good  evidence  for  believing  that,  when  some 
substances  are  dissolved  in  certain  liquids,  especially  in- 
organic salts  and  acids  in  water,  they  are  partially  disso- 
ciated ;  but  this  effect  is  complicated  by  certain  electrical 
phenomena  to  be  discussed  later. 

Application  of  Principle  of  Stable  Equilibrium.  The  effect 
of  changing  the  temperature  may  be  determined  from  the 
"  principle  of  stable  equilibrium."  If  the  combination 
involves  a  development  of  heat-energy,  then  adding  heat- 
energy  will  produce  dissociation.  If,  on  the  contrary,  com- 
bination is  accompanied  by  an  absorption  of  heat-energy, 
then  taking  away  energy  will  produce  dissociation  or 
breaking  up  of  the  compound ;  and  the  addition  of  heat- 
energy  must  produce  combination. 

SOLUTION 

203.  Whenever  any  two  substances  are  placed  in  con- 
tact, solid,  liquid,  or  gaseous,  there  is  no  reason  for  not 
thinking  that  each  dissolves  in  the  other  somewhat ; 
that  is,  that  some  molecules  of  each  substance  become 


244  THEORY  OF  PHYSICS  [CH.  IV 

detached,  as  it  were,  and  interpenetrate.  This  is  known 
to  occur  in  many  cases ;  e.  g.  salt  and  water,  alcohol  and 
water,  mercury  and  tin,  etc.  In  all  cases  there  are  energy- 
changes,  and  so  heat-effects,  which  can  be  measured. 
Sometimes  energy  is  absorbed  from  surrounding  bodies  ;  at 
other  times  it  is  given  out  to  them. 

No  process  of  solution  proceeds  indefinitely.  For  a 
given  temperature  there  is  a  definite  state  of  saturation. 
The  effect  of  raising  the  temperature  may  be  determined 
from  the  "  principle  of  stable  equilibrium."  The  saturated 
solution  is  in  stable  equilibrium.  Imagine  a  further  solu- 
tion ;  and  let  the  solution  be  one  which  involves  a  develop- 
ment of  heat-energy.  Hence,  if  heat-energy  is  added  to 
this  saturated  solution,  the  solution  must  be  resolved  into 
its  constituents;  it  becomes  supersaturated.  If,  on  the 
other  hand,  the  solution  was  one  which  is  accompanied  by 
absorption  of  heat-energy,  then  adding  heat-energy  to  the 
saturated  solution  would  make  the  solution  unsaturated, 
and  more  solution  will  take  place  if  possible. 

In  some  solutions,  as  noted  above,  there  is  evidence  that 
the  molecules  which  go  into  solution  are  dissociated.  It 
will  be  shown  later  on  that  there  is  reason  for  believing 
that  this  is  the  case  with  all  liquids  which  can  conduct 
electricity.  No  pure  liquid  is  a  conductor,  only  certain 
solutions ;  and  these  solutions  are  probably  dissociated 
molecules  dissolved  in  the  pure  liquid. 


CHAPTER  V 

TRANSFER   OF   HEAT-ENERGY 

As  explained  in  mechanics,  there  are  two  ways,  in  gen- 
eral, in  which  energy  may  be  transferred  from  one  point 
to  another.  One  is  by  the  energy  being  carried  by  a  por- 
tion of  matter ;  e.  g.  a  cannon-ball  conveys  energy  from  the 
exploded  powder  in  the  cannon  to  the  target  or  obstacle 
struck.  The  other  is  by  waves  ;  e.  g.  sound-waves  convey 
energy  from  the  sounding  body  to  the  ear. 

There  are  illustrations  of  both  these  methods  in  the 
transfer  of  heat-energy,  i.  e.  of  energy  from  the  molecules 
of  one  body  to  those  of  another.  Three  general  processes 
are  distinguished :  Convection,  Conduction,  Radiation. 

204.  Convection.  In  this  process  some  portion  of  the 
body  whose  temperature  is  raised  moves  away  to  other 
places  in  the  body  where  the  temperature  is  lower,  thus 
tending  to  raise  the  temperature  throughout.  Portions  of 
solids  cannot  move  about ;  so  convection  is  limited  to 
fluids,  i.  e.  to  liquids  and  gases.  If  a  vessel  of  any  liquid 
is  placed  over  a  fire,  the  portions  of  the  liquid  near  the 
bottom  become  hotter  and  so  expand;  but  on  expansion 
the  portions  become  less  dense,  and  so  tend  to  move  up- 
ward towards  the  top  of  the  liquid.  Thus  the  energy  is 
transferred.  If  the  liquid  was  heated  on  top,  there  would 
be  no  convection.  Similarly,  there  can  be  convection  in  a 
gas,  if  it  is  heated  from  below. 

The  process  of  convection  is  of  fundamental  importance 
in  draughts  of  chimneys,  ventilation,  and  in  the  great 
phenomena  of  nature,  such  as  winds  and  ocean-currents. 


246 


THEORY  OF  PHYSICS 


[CH.  V 


205.  Conduction.  In  conduction  the  separate  portions  of 
matter  do  not  move  bodily  and  thus  transfer  the  energy ; 
but  they  hand  it  on  from  one  portion  to  the  other.  Thus, 
if  a  poker  has  one  end  in  a  fire,  that  end  has  its  tempera- 
ture raised ;  and  this  in  turn  raises  the  temperature  of  the 
portion  next  it,  and  so  on  down  the  entire  length  of  the 
poker.  The  other  end  finally  has  its  temperature  raised, 
but  not  until  all  the  intervening  portions  have  had  their 
temperatures  also  raised.  It  is  found  by  experiment  that 
all  solids  which  can  conduct  electricity,  i.  e.  all  metals,  are 
good  heat-conductors  ;  other  solids  are  not.  Of  the  liquids 
mercury  is  the  only  good  conductor  ;  and  all  gases  are  poor 
conductors.  (To  study  conduction  in  fluids,  the  heat- 
energy  must,  of  course,  be  supplied  from  above,  so  as  to 
avoid  convection.) 

There  are  many  illustrations  of  the  power  of  metals  to 
conduct  away  heat-energy.  One  of  the  most  interesting  is 
the  ordinary  miner's  safety-lamp,  in  which  the  flame  is 
surrounded  by  metal-gauze  and  s.olid  metal  pieces,  so  that 
the  heat-energy  from  the  flame  is  conducted  away  by  the 
metal ;  and  thus  the  temperature  of  the  escaping  gases  is 
too  low  to  ignite  outside  the  lamp. 

The  following  table  gives  an  approximate  idea  of  the 
conducting  power  of  different  bodies  :  — 

TABLE   XI 
THERMAL  CONDUCTIVITIES 


Copper    .     .     . 

.96 

Glass      .     .   ... 

.0005 

Iron  .... 

.20 

Wool      .     .     . 

.00012 

Stone      .     .     . 

.006 

Paper      .     .     . 

.000094 

Water     .     .     . 

.002 

Air    .     .     .     . 

.000049 

It  should  be  noted  that  the  conducting  power  of  any  sub- 
stance is  different  for  different  temperatures. 


206J  TRANSFER  OF  HEAT-ENERGY  247 

206.  Radiation.  This  is  the  process  by  which  the  heat- 
energy  is  transferred  from  one  point  to  another  by  means 
of  waves.  The  existence  of  these  waves  may  be  demon- 
strated by  the  usual  tests  of  wave-motion,  which  will  be 
explained  later  under  the  section  LIGHT  ;  they  advance 
with  a  finite  velocity ;  they  can  interfere ;  they  can  be 
diffracted.  Further,  they  can  be  polarized;  and  this,  as 
we  shall  see,  is  proof  that  the  waves  are  transverse. 
These  waves,  too,  are  not  in  ordinary  matter,  as  are 
sound-waves,  but  can  be  propagated  across  a  vacuum,  or 
from  the  sun  and  stars  to  the  earth.  Since  waves  must  be 
in  some  medium,  this  proves  that  there  is  a  medium  which 
permeates  all  spaces,  large  and  small,  between  portions  of 
ordinary  matter.  This  medium,  whose  existence  is  thus 
demonstrated,  is  called  "  the  ether."  It  must  have  inertia, 
because  a  finite  time  is  taken  for  the  propagation  of  the 
waves  across  a  finite  space.  But  the  other  properties  of 
the  ether  are  quite  unknown. 

There  are,  then,  waves  in  the  ether  which  carry  energy 
away  from  various  sources  ;  and  their  properties  are  now 
well  known.  They  are  transverse  ;  they  travel  through  the 
ether  with  a  definite  velocity  which  is  independent  of  the 
wave-length  ;  but  when  portions  of  ordinary  matter  are  im- 
mersed in  the  ether  the  velocity  of  the  waves  is  changed, 
and  is  changed  differently  for  different  waves ;  the  wave- 
lengths may  be  measured  by  suitable  means ;  and  their  in- 
tensity may  be  determined.  When  these  waves  reach  any 
obstacle,  there  will  be  reflected  and  refracted  waves  in  gen- 
eral, if  the  obstacle  is  large  compared  with  the  length  of  the 
waves ;  and  the  laws  of  the  reflected  and  refracted  waves 
will  be  discussed  later  in  LIGHT.  It  is  sufficient  to  say 
here  that  all  these  waves  obey  the  same  laws  as  so-called 
light-waves.  (See  below.)  Some  of  the  energy  will  also 
be  given  the  obstacle,  i.  e.  will  be  "  absorbed  "  by  it.  The 
effect  of  this  absorption  on  the  obstacle  depends  upon 
what  work  is  done  there  as  a  result  of  the  energy  being 


248  THEORY  OF  PHYSICS  [CH.  V 

received  from  the  waves.  It  is  found  by  experiment  that, 
if  the  obstacle  is  any  ordinary  body,  the  absorbed  energy 
in  general  produces  heat-effects.  But,  if  the  waves  have 
lengths  within  certain  limits,  and  if  they  fall  on  the  eye, 
some  of  the  energy  is  used  in  producing  the  sensation 
Light.  Again,  if  the  obstacle  is  one  of  many  chemical 
substances,  e.  g.  a  photographic  film,  some  of  the  energy 
goes  to  causing  certain  chemical  reactions.  If  the  waves 
are  longer  than  any  of  these,  their  energy  may  be  con- 
sumed in  producing  electrical  oscillations  in  conductors. 

Ether-waves,  then,  produce  heat-effects  when  they  are  ab- 
sorbed, only  if  they  are  absorbed  by  special  bodies.  Since 
in  some  cases  they  produce  the  sensation  light,  this  fact 
renders  easy  the  study  of  their  laws.  As  will  be  proved 
later,  it  is  known  that  the  velocity  of  these  waves  through 
the  pure  ether  is  almost  exactly  3  X  1010  cm.  per  sec.  Fur- 
ther, those  waves  which  produce  light-sensation  have  wave- 
lengths lying  roughly  between  0.00004  and  0.00008  cm.,  i.  e. 
are  about  ^Jnnj-tli  of  an  inch  long.  Waves  as  short  as 
0.00001  cm.  and  as  long  as  0.002  cm.  have  been  measured. 
The  energy  carried  by  the  waves,  or  their  intensity,  may  best 
be  studied  by  letting  waves  of  different  lengths  be  ab- 
sorbed by  some  body,  such  that  all  the  energy  goes  into- 
heat-effects.  By  suitable  means  the  entire  series  of  trains 
of  waves  emitted  from  any  source  may  be  analyzed  into 
separate  trains  of  waves,  each  train  having  a  definite  wave- 
length which  differs  minutely  from  those  of  its  neighbors. 
Such  an  analysis  may  be  done  by  a  prism  or  a  grating 
(see  LIGHT).  If,  then,  a  curve  is  drawn,  the  locus  of  points 
whose  horizontal  distances  from  some  fixed  line  give  wave- 
lengths, and  whose  vertical  distances  from  another  fixed 
line  perpendicular  to  the  first  give  intensities,  it  is  called 
the  "  energy-curve  "  for  the  particular  source.  Certain  in- 
teresting laws  have  been  observed  between  energy-curves 
and  temperatures  of  the  source.  Thus,  the  higher  the  tem- 
perature of  the  source,  so  much  further  does  the  energy- 


206]  TRANSFER  OF  HEAT-ENERGY  249 

E 


ENERGY  CURVE  * 

FlG.    144. 

curve  extend  towards  the  left ;  that  IB,  there  are  now, 
among  the  waves  emitted,  some  shorter  than  there  were 
before.  At  the  same  time,  as  the  temperature  is  raised,  the 
intensity  of  each  particular  train  of  waves  is  increased ;  or 
the  energy-curve  is  raised,  —  not  uniformly,  however. 

The  energy  in  these  ether-waves  is  emitted  by  different 
forms  of  matter.  In  fact  every  piece  of  matter  in  the  uni- 
verse is  undoubtedly  producing  ether-waves.  A  solid  or 
a  liquid  is  sending  out  what  is  called  a  "  continuous  spec- 
trum ; "  that  is,  in  the  series  of  waves  emitted,  there  are 
trains  of  all  possible  wave-lengths  in  between  certain 
limits,  there  are  no  special  waves  absent.  A  gas  or  vapor, 
on  the  other  hand,  produces  a  "  discontinuous  "  spectrum  ; 
certain  waves  are  present,  others  are  absent.  The  ones 
which  are  present  depend  upon  the  condition  and  nature 
of  the  gas  or  vapor. 

Few  bodies  absorb  practically  all  the  waves  which  fall 
upon  them  ;  most  of  them  absorb  only  certain  definite 
waves.  And,  further,  the  waves  which  are  absorbed  and 
the  intensity  of  the  absorption  depend  upon  the  tempera- 
ture of  the  absorbing  body.  There  must  be  some  con- 
nection between  the  radiating  power  of  a  body  and  its 
absorptive  power.  For,  consider  a  body  enclosed  in  a  space, 


250  THEORY  OF  PHYSICS  [CH.  V 

e.  g.  a  hollow  sphere,  whose  bounding  surface  is  kept  at  a 
constant  temperature,  and  which  is  lined  with  some  sub- 
stance having  the  power  to  emit  all  wave-lengths  up  to  a 
certain  limit.  Some  of  these  waves  are  reflected  from  the 
body,  some  are  transmitted,  some  are  absorbed. '  But  some 
waves  are  also  emitted  by  the  body  itself ;  and,  since  ex- 
periments prove  that  the  temperature  of  the  body  inserted 
gradually  becomes  equal  to  that  of  the  enclosing  walls 
and  then  does  not  change,  the  waves  emitted  when  this 
temperature  is  reached  must  be  identical  in  every  respect 
with  those  absorbed.  In  other  words,  the  emissive  power 
of  any  body  at  any  temperature  exactly  equals  the  ab- 
sorptive power  at  that  same  temperature.  Thus,  red  glass 
is  called  so,  because  it  absorbs  all  the  colors  which  come 
from  any  hot  source,  except  red.  But  if  the  glass  is  raised 
to  that  temperature,  it  will  emit  those  waves  which  it 
absorbed  before,  viz.  bluish  green,  as  will  be  apparent  if  it 
is  carried  while  hot  into  a  dark  room.  Sodium  vapor  at 
a  high  temperature  has  the  power  of  emitting  a  few  char- 
acteristic waves,  especially  two  which  produce  the  sensa- 
tion yellow.  So,  if  white  light,  i.  e.  waves  from  a  very 
hot  solid,  is  passed  through  sodium  vapor  at  a  low  tem- 
perature, these  particular  waves  are  absorbed  and  are 
therefore  absent  in  the  transmitted  waves.  This  is,  of 
course,  a  general  mechanical  principle,  which  has  been 
illustrated  before  in  the  case  of  two  tuning-forks  (see  Art. 
129).  If  a  fork  is  set  in  vibration,  the  waves  produced 
falling  upon  another  fork  of  the  same  frequency  will  set  it 
in  vibration.  That  is,  the  second  fork  absorbs  the  energy 
from  those  waves  which  it  itself  could  produce,  because  it 
has  the  same  frequency. 

207.  Reflection,  Emission,  Absorption,  Transmission.  A 
body  which  absorbs  well  can  also  emit  energy  well ;  as  is 
shown  in  all  blackened  bodies  which  are  not  polished. 
But,  when  energy,  in  the  form  of  waves,  falls  upon  a  body, 
part  is  reflected,  and  the  rest  absorbed  or  transmitted ;  so 


208]  TRANSFER  OF  HEAT-ENERGY  251 

that,  if  a  body  reflects  well,  it  must  absorb  poorly.  Thus, 
polished  surfaces,  particularly  metals,  must  absorb  very 
little  energy,  and  so  must  emit  very  little,  because  they  are 
such  good  reflectors. 

Again,  the  amount  of  energy  which  is  transmitted  de- 
pends upon  the  amount  reflected  and  absorbed.  An  equa- 
tion may  be  written :  Incident  energy  =  reflected  +  ab- 
sorbed +  transmitted. 

The  question  of  the  transparency  of  any  body  to  these 
ether-waves  depends  largely  on  their  wave-length  (or, 
better,  their  wave-number)  and  their  intensity.  Many 
substances,  transparent  to  visible  waves,  are  almost  en- 
tirely opaque  to  longer  or  shorter  waves.  Thus,  glass 
transmits  freely  light- waves,  but  shuts  off  the  others.  A 
solution  of  iodine  in  carbon  bisulphide  is  entirely  opaque 
to  light-waves,  but  is  transparent  to  the  longer  waves. 
And  in  every  case,  the  more  intense  the  waves,  i.  e. 
the  higher  the  temperature  of  the  source,  so  much  the 
greater  are  the  chances  of  any  wave  getting  through 
any  substance. 

208.  Flow  of  Heat-Energy.  In  Convection  and  Conduc- 
tion it  is  obvious  that  heat-energy  is  carried  from  places  of 
high  temperature  to  places  of  low ;  and  the  rapidity  of 
this  flow  varies  directly  as  the  difference  in  temperature 
of  the  two  places. 

The  same  statements  are  also  true  of  Eadiation.  If  a 
body  at  a  low  temperature  is  placed  inside  a  second  body 
which  is  so  made  as  to  emit  no  waves  from  the  outer  surface, 
and  which  is  at  a  higher  temperature  than  the  first  body, 
each  radiates  a  certain  amount  of  energy  depending  only  on 
its  own  temperature  and  condition ;  each  body  also  absorbs 
a  certain  amount  of  the  energy  radiated  by  the  other.  The 
hot  body,  though,  radiates  more  energy  than  it  absorbs  ; 
while  the  opposite  is  true  of  the  cold  body.  Consequently, 
the  temperature  of  the  hot  body  falls,  and  that  of  the  cold 
body  rises,  until  the  temperatures  of  the  two  are  the  same. 


252  THEORY  OF  PHYSICS  [CH.  V 

Thus,  heat-energy  has  passed  from  a  body  at  a  high  tem- 
perature to  a  body  at  a  lower  one ;  and  the  rapidity  of  the 
process  must  vary  directly  as  the  difference  in  temperature. 

If  there  is  no  difference  of  temperature,  there  is  no  flow 
of  heat-energy ;  and,  if  there  is  a  difference,  the  flow  is  al- 
ways from  high  to  low  temperature. 

The  analogy  should  be  noticed  to  the  flow  of  a  fluid 
from  places  of  high  pressure  to  places  of  low.  For  in- 
stance, water  flows  from  a  high  tank  to  a  lower  one ;  a 
gas  compressed  into  a  vessel  will  expand,  if  it  can,  into 
another  vessel  where  the  pressure  is  less.  And  the  rate  of 
the  flow  varies  directly  as  the  difference  in  pressure.  At 
a  great  pressure  the  fluid  has  more  energy  than  at  a  lower 
one;  so,  in  the  flow  of  the  fluid,  what  may  be  called 
"volume-energy"  flows  from  high  to  low  pressure. 


CHAPTEK  VI 

KINETIC  THEORY   OF  MATTER 

209.  IT  is  impossible  to  understand  how  waves  can  be 
produced  in  the  ether  by  means  of  ordinary  matter,  or 
how  the  waves  can  convey  heat-energy  from  one  portion 
of  matter  to  another,  unless  there  is  some  connection  be- 
tween matter  and  the  ether,  and  unless  the  smallest  por- 
tions   of    matter    are    in   motion.     Waves   in   the    ether 
must  therefore  be  considered. as  produced  by  the  vibrations 
of  extremely  small  portions  of  matter.     Consequently  all 
portions  of  matter,  however  small,  are  considered  as  being 
in  motion ;  the  molecules  are  thought  to  be  moving  bodily ; 
and  the  parts  of  the  molecules,  if  there  are  any,  may  be 
making  vibrations. 

210.  States  of  Matter :  Solids.    The  idea  of  a  solid  is,  then, 
a  fixed  configuration  of  molecules,   where  the  molecules 
may  vibrate  about   positions  of   equilibrium   but   cannot 
move  from  one  point  to  another  in  the  solid.     The  reason 
why  a  solid  emits  a  continuous  spectrum  is  because  the 
vibrations  of  the  portions  of  the  molecules  are  so  ham- 
pered by  the  close  proximity  of  other  molecules  that  they 
vibrate  in  all  periods  within  certain  limits,  and  so  produce 
waves  of  all  lengths.     The  same  explanation  applies  to 
the  waves  emitted  by  a  liquid. 

Liquids.  In  a  liquid  the  molecules  have  increased  free- 
dom of  motion,  and  can  move  about  from  one  point  to 
another.  They  undoubtedly  are  moving  with  considerable 
velocity ;  and  so  it  is  easy  to  understand  how  some  may 
escape  from  the  surface  and  thus  cause  evaporation. 


254 


THEORY  OF  PHYSICS 


[CH.  VI 


Vapors.  In  a  vapor  or  gas  the  molecules  are  no  longer 
close  together ;  and  they  are  moving  with  great  velocity  in 
all  directions.  In  the  vapor  over  a  liquid,  some  of  the 
molecules  in  their  motion  will  undoubtedly  strike  the  sur- 
face of  the  liquid,  and  become  entangled.  There  is,  thus, 
a  process  of  continuous  evaporation  and  condensation ;  and 
the  state  of  saturated  vapor  is  reached  when  the  amount 
evaporated  in  a  definite  time  equals  that  condensed  in  the 
same  time.  Similarly,  in  dissociation  there  is  undoubtedly 
a  state  of  continuous  dissociation  and  combination  existing ; 
and  equilibrium  is  reached  when  the  two  opposite  pro- 
cesses are  equal. 

Gases.  In  a  gas  the  molecules  move  freely  in  the  inter- 
vals of  time  between  collisions  with  each  other  or  with 
the  walls  of  the  vessel ;  and  so  they  have  what  is  called 
"  a  free  path."  During  their  passage  along  their  free  paths, 
the  molecules  are  uninfluenced  by  each  other ;  and  so  the 
portions  of  the  molecules  may  by  their 
vibrations  send  out  waves  of  certain 
definite  wave-numbers.  It  is  easy,  too, 
to  understand  conduction  of  heat-energy, 
diffusion,  and  other  properties  of  gases 
on  this  kinetic  theory. 

If  a  closed  space  is  so  thoroughly 
exhausted  by  means  of  air-pumps  as  to 
allow  the  molecules  of  a  gas  to  have 
free  paths  of  several  inches,  the  entire 
nature  of  the  gas  changes,  and  many 
new  properties  are  observed.  This  con- 
dition is  sometimes  called  a  fourth  state 
of  matter.  One  of  the  most  interesting 
properties  of  this  state  is  that  there  are 
FIG  145  curious  motions  produced  in  solid  bodies 

placed  in  the  exhausted  space,  when 
ether-waves  fall  upon  them.  If  a  vane,  made  of  two  arms 
carrying  at  each  end  a  plate  of  mica  polished  on  one  side 


211]  KINETIC   THEORY   OF  MATTER  255 

and  blackened  on  the  other,  is  balanced  on  an  axis  in  a 
glass  bulb  which  is  thus  exhausted,  it  will  revolve  when 
any  hot  object  is  brought  near,  the  blackened  sides  of  the 
mica  always  moving  away  from  the  object.  The  explana- 
tion is  not  difficult.  The  blackened  surface  absorbs  the 
energy  from  the  ether-waves ;  its  temperature  is  raised ; 
and  so  the  molecules,  as  they  rebound  from  the  surface, 
receive  an  extra  velocity.  Owing  to  the  reaction  of  the 
molecules  on  the  mica  surface,  the  latter  is  driven  back, 
away  from  the  hot  body  emitting  the  waves.  This  phe- 
nomenon would  not  take  place  in  an  ordinary  gas,  because 
any  rise  in  temperature  or  increased  velocity  of  some  of 
the  molecules  would  be  almost  instantly  communicated  to 
the  rest  of  the  gas ;  and  so  there  would  be  an  equal  force 
on  both  sides  of  the  mica-vanes. 

211.  Kinetic  Theory  of  Gases.  There  is  a  definite  theory 
of  gases,  more  specific  than  the  general  one  given  above ; 
and  it  states  that  it  is  possible  to  prove  that  a  collec- 
tion of  small,  hard,  smooth,  elastic  spheres,  identically 
alike,  thrown  at  random  into  a  large  space  with  rigid 
walls  will  have  certain  properties  identical  with  those  of 
perfect  gases.  It  is  necessary,  further,  to  assume  that  the 
space  is  so  large  and  the  spheres  so  numerous  that  the 
"  principle  of  statistics  "  may  be  applied.  That  is,  that,  al- 
though the  properties  of  an  individual  sphere  may  change 
each  instant,  the  average  for  all  the  spheres  of  the  same 
property  (e.  g.  speed)  will  not  change  unless  the  conditions 
are  changed. 

a.  Energy  and  Temperature.  A  set  of  spheres  like  the 
one  described  consists  of  a  great  number,  each  one  having 
the  same  mass,  and  each  one  moving  with  a  definite  veloc- 
ity at  any  instant.  The  velocities  of  the  different  spheres 
vary  widely ;  and  that  of  any  individual  sphere  changes  at 
each  collision.  But  at  any  instant  there  is  a  certain  aver- 
age velocity  in  any  direction ;  and  there  is  also  a  certain 
average  kinetic  energy.  Since  the  particles  are  perfectly 


256  THEORY  OF  PHYSICS  [CH.  VI 

smooth  and  elastic,  the  entire  kinetic  energy  of  trans- 
lation is  not  changed  by  any  collisions  ;  and  so,  if  the  walls 
of  the  vessel  are  rigid,  the  average  kinetic  energy  cannot 
change.  But,  if  the  walls  are  free  to  move  out,  they  will 
do  so  under  the  pressure  produced  by  the  impacts  of  the 
spheres,  external  work  will  be  done ;  and  so  the  average 
kinetic  energy  of  translation  will  decrease.  Conversely,  if 
the  walls  are  forced  in  by  any  external  force,  the  average 
kinetic  energy  of  translation  is  increased.  These  proper- 
ties of  the  average  kinetic  energy  of  translation  of  the  set 
of  spheres  correspond  perfectly  to  those  of  the  tempera- 
ture of  an  ordinary  gas.  Further,  it  may  be  proved  mathe- 
matically that  two  sets  of  spheres,  thoroughly  mixed,  will 
be  in  equilibrium  when  the  average  kinetic  energies  of 
translation  of  the  two  sets  are  equal.  That  is :  if  mi  is 
the  mass  of  any  one  sphere  of  the  one  set,  and  u\  is  the 
average  velocity  of  that  set,  while  m2  and  u2  correspond  to 
the  second  set,  there  will  be  equilibrium  when 

mi  Uiz  =  ra2  u22 (1) 

And  two  gases  are  in  equilibrium  if  their  temperatures 
are  the  same ;  so,  on  the  kinetic  theory,  this  equation  may 
be  considered  as  the  mathematical  condition  for  equality 
of  temperature  of  two  gases. 

b.  Momentum  and  Pressure.  As  the  spheres  strike  the 
walls  of  the  vessel,  their  velocity  perpendicular  to  the 
walls  is  reversed,  and  so  there  is  a  change  in  momen- 
tum. But,  whenever  there  is  a  change  in  momentum, 
there  is  a  force  produced ;  so  the  spheres  exert  a  force 
on  the  walls,  normally  outward.  It  is  not  difficult  to 
calculate  this  force.  Consider  a  cubical  vessel  containing 
the  set  of  spheres  ;  and  let  each  edge  have  a  length  a. 
Since  the  motion  of  the  spheres  is  entirely  a  random  one, 
there  is  as  much  momentum  parallel  to  one  edge  of  the 
cube  as  to  any  other.  So  the  entire  set  of  spheres  may 
be  considered  as  divided  into  three  equal  sets,  each  mov- 


211]  KINETIC  THEORY  OF  MATTER  257 

ing  parallel  to  one  edge  with  a  certain  average  velocity. 
Thus,  let 

m  =  mass  of  each  sphere, 

n  =  number  of  spheres  in  1  cubic  centimetre, 

u  =  average  velocity  parallel  to  one  edge. 

Hence  there  are  n  a5  spheres  in  the  cube,  and  J  n  aB  are 
moving  parallel  to  each  edge.  These  strike  the  end  side 
with  a  momentum  J  n  oft .  m  u ;  but  this  is  reversed  at  the 
wall,  that  is,  the  velocity  becomes  —  u ;  and  so  the  change  in 
the  momentum  is  f  n  a3 .  m  u.  It  takes  each  sphere,  moving 

with  the  velocity  u,  a  time  —  for  it  to  pass  over  to  the 
opposite  wall  and  back  again ;  or,  in  other  words,  each 
sphere  returns  and  collides  with  the  same  wall  —  times 

in  one  second.  Consequently  the  entire  change  of  momen- 
tum at  the  wall  in  one  second  is 

4  n  as  m  u  •  — —  =  \  m  n  u2  a2. 
2a 

By  the  definition  of  force  (Art.  30)  this  is  the  force  acting 
on  the  wall  whose  area  is  a2.  Hence  the  pressure,  the 
force  per  unit  area,  is 

p  =  ^mnu2 (2) 

m  is  the  mass  of  each  sphere,  n  is  the  number  of  spheres  in 
1  cc. ;  hence  m  n  is  the  mass  in  1  cc.,  i.  e.  the  density,  p. 
So 

P  =  t«*P (2  a) 

But  since,  if  the  average  kinetic  energy  of  translation  is 
constant,  u2  is  also,  this  formula  may  be  read :  If  the  aver- 
age kinetic  energy  of 'translation  of  a  set  of  spheres  is  con- 
stant, the  pressure  on  the  walls  is  proportional  to  the 
density.  This  is  identical  with  Boyle's  laws  for  perfect 


258  THEORY  OF   PHYSICS  [CH.  VI 

Further,  assuming  this  law  to  be  rigidly  true  for  actual 
gases,  the  average  velocity  of  a  molecule  of  a  gas  may  be 
calculated  for  any  pressure.  Substitute  for  p  its  value, 
and  for  p  its  value  for  the  particular  value  of  p  ;  and  then 


If  there  are  several  sets  of  spheres  enclosed  in  the  same 
space,  each  will  obviously  produce  its  own  pressure  on  the 
walls  of  the  vessel,  provided  that  the  space  is  large  enough 
and-  that  one  set  does  not  vastly  outnumber  any  other. 
This  is  Dalton's  Law.  Thus,  the  entire  pressure 

P  =  pi  +  pz  +  ps  +  etc., 
where  p\  —  \  m\  %  Ui2, 

p2  =  ^  mz  nz  Uz2,         etc. 


If  there  is  equilibrium  of  the   sets  of   spheres,  equation 
(1)  requires  that 


wi  Ui2  =  ??i2  u22  =  ms  us2  =  etc.  =  c  (say). 
Hence  P  =  %  c  (HI  +  nz  +  ns  +  etc.). 

That  is,  the  pressure  in  a  mixture  of  gases,  at  a  given  tem- 
perature depends  simply  upon  the  total  number  of  particles, 
not  in  the  least  upon  the  mass  of  any  individual  com- 
ponent. (This  statement  may  be  regarded  as  an  extension 
.of  Avogadro's  Law,  which  is  given  below.) 

c.  Expansion.     It  may  be  shown,  too,  that  Charles'  Law 
is  true  for  sets  of  spheres.     That  is,  if  the  volume  is  kept 
constant,  but  the  kinetic  energy  increased,  the  pressure 
will  increase  at  a  rate  which  is  the  same  for  all  sets  of 
spheres. 

d.  Avogadro's  Law.     If  there  are  two  sets  of  spheres,  at 
the  same  temperature  and  pressure,  equations  (1)  and  (2) 

give 

mi  uf  =  mz  u£ 

mi  ni  u-?  —  mz  nz  uz2. 
And  so  ni  =  nz   .....     ...     (3) 


211]  KINETIC   THEORY   OF  MATTER  259 

or,  the  number  of  spheres  of  the  first  set  in  1  cc.  is  the 
same  as  the  number  of  spheres  of  the  second  set  in  an 
equal  volume. 

Applied  to  gases,  this  principle  is  called  Avogadro's  Law  ; 
and  it  states  that,  if  two  gases  are  under  the  same  con- 
ditions of  temperature  and  pressure,  they  both  have  the 
same  number  of  molecules  (or  smallest  parts)  in  equal 
volumes. 

The  density  of  a  gas,  p,  equals  m  n  ;  and  so  the  ratio  of 
the  densities  of  two  gases  at  the  same  temperature  and 
pressure  is 

pl  I  p2  =  mi  n  /  m2  n  =  mi  /  w2     .     .     .     (4) 

mi  is  the  actual  mass  of  a  single  molecule  ;  and  this  for- 
mula gives  a  method  for  comparing  the  masses  of  molecules 
of  different  gases,  because  the  densities  are  measurable 
quantities. 

What  is  called  the  "  molecular  weight  "  of  a  gas  in 
Chemistry  is  a  number  which  is  proportional  to  the  mass 
of  a  single  molecule  of  the  gas,  the  factor  of  proportional- 
ity being  so  chosen  that  the  molecular  weight  of  hydrogen 
is  2.  So  the  molecular  weights  of  two  gases,  w\  and  w2, 
must  be  in  the  same  ratio  as  ml  and  m2  ;  and  there  will  be 
the  same  number,  N,  of  molecules  of  any  gas  in  a  mass  of 
that  gas  equal  to  its  molecular  weight. 

Boyle's  and  Charles'  laws  for  a  gas  give 


where  M  is  the  mass  occupying  a  volume  v.  Let  this 
mass  be  wt  that  is,  a  number  of  grams  which  equals  the 
molecular  weight  ;  there  are  then  N  molecules  in  the 
space  v.  Consider  a  second  gas  whose  equation  is 

'  V' 


260  THEORY  OF  PHYSICS  [CH.  VI 

and  take  wr  grams,  where  w'  is  its  molecular  weight. 
There  are  N  molecules  of  this  second  gas  in  the  volume 
0'.  But  equal  numbers  of  molecules  of  different  gases 
occupy  equal  volumes  if  temperature  and  pressure  are  the 
same.  Hence,  when 

M=  w,  M'  =  wr,     vr  =  v  if  p  =p'  and  T  =  Tf. 
Or  Rw  =  Rw'  =  RQ,  ......     (5) 

where  RQ  is  a  constant  for  all  gases.  So,  if  M  =  q  w,  that 
is,  if  the  number  of  grams  of  the  gas  is  equal  to  q  times  the 
molecular  weight,  the  general  law  for  a  gas  becomes 


Writing  v/  q  =  VQ,  or  the  volume  which  would  be  occupied 
by  a  mass  of  the  gas  equal  to  the  molecular  weight,  this 
becomes 


/A\ 

(6) 


Avogadro's  law  (or  hypothesis)  is  the  basis  of  many 
chemical  theories  ;  and,  although  not  rigidly  proved  for 
actual  gases,  it  must  be  reasonably  correct.  It  is  observed 
that,  when  equal  volumes  of  hydrogen  and  chlorine  gas 
combine  to  form  hydrochloric  acid  gas,  the  volume  formed 
is  the  sum  of  the  two  volumes  of  the  constituents,  if  the 
pressure  and  temperature  are  kept  constant.  The  chemical 
formula,  as  ordinarily  written,  is 

H  +  Cl  -  HC1. 

On  its  face,  this  would  mean  that  one  molecule  was  formed 
from  two  ;  but  this,  on  Avogadro's  hypothesis,  would 
require  that  the  resulting  volume  of  HC1  should  be  one- 
half  that  of  the  sum  of  the  two  component  volumes,  if 
pressure  and  temperature  are  the  same.  This  discrepancy 
between  hypothesis  and  experiment  may  be  explained  if  it 


211]  KINETIC  THEORY  OF  MATTER  261 

is  assumed  that  a  molecule  of  H  has  two  atoms,  and  one  of 
Cl  has  two  atoms  ;  then  HC1  is  a  compound  of  an  atom 
of  H  and  an  atom  of  Cl.  Thus,  if  there  are  n  molecules 
of  H  in  a  volume  v,  there  are  also  n  molecules  of  Cl  in  an 
equal  volume,  if  temperature  and  pressure  are  the  same. 
On  combination,  as  expressed  in  the  formula 

Ha  +  Cla  =  2HCl, 

2  n  molecules  HC1  are  formed  ;  and  so  twice  the  volume  is 
necessary  to  keep  the  same  temperature  and  pressure. 

Similarly,  when  water-vapor  is  formed,  2  n  molecules  of 
H  combine  with  n  molecules  of  0  to  form  2  n  molecules 
H20.  So  the  steam  must  be  compressed  into  a  volume 
which  is  twice  the  volume  v,  if  the  pressure  and  tempera- 
ture are  to  be  kept  unchanged.  That  is,  there  is  a  "  con- 
densation "  from  volume  3  v  (2  v  for  H  and  v  for  0)  to 
volume  2  v. 

In  general,  when  any  compound  gas  is  formed  from  other 
gases,  the  volumes  bear  simple  mathematical  relations  to 
each  other,  if  the  temperature  and  pressure  are  the  same 
for  all  the  gases ;  and  this  leads,  on  Avogadro's  hypothesis, 
to  a  definite  idea  of  the  nature  of  the  molecule. 


CHAPTEK  VII 

THERMODYNAMICS 

THERMODYNAMICS  is  the  name  given  to  the  science  which 
studies  the  application  of  mechanical  principles  and  equa- 
tions to  the  physical  properties  of  heat-energy. 

212.  First  Principle  of  Thermodynamics.  This  is  nothing 
but  the  application  of  the  principle  of  the  conservation  of 
energy  to  heat-effects.  It  asserts  that,  whenever  energy  is 
spent  in  producing  heat-effects,  the  amount  of  work  done 
exactly  equals  the  sum  of  the  internal  and  the  external 
work  (see  Art.  167).  This  is  perfectly  in  accord  with  all 
experiments.  The  amount  of  work  required  to  produce 
any  "  effect  of  heat  "  is  the  same,  no  matter  how  the  work 
is  done.  Thus,  the  amount  of  energy  necessary  to  raise 
the  temperature  of  1  gram  of  water  from  10°  to  11°  C.  is 
the  same  if  the  work  is  done  by  turning  a  paddle  in  the 
water,  or  if  the  energy  is  produced  by  heating  a  wire  by 
means  of  an  electric  current.  This  amount  of  work,  called 
J,  the  "  mechanical  equivalent  of  Heat,"  has,  as  has  been 
said,  the  value  4.2  X  107.  Further,  J,  as  found  by  means 
of  the  equation 


(see  Art.  182)  is  the  same,  within  the  limits  of  accuracy  of 
the  experimental  determination  of  CP)  C,,  and  E. 

213.  Second  Principle  of  Thermodynamics.  This  principle 
asserts  what  must  be  regarded  as  an  axiom,  viz.  that  heat- 
energy  of  itself  cannot  pass  from  one  body  to  another  if 
the  first  body  is  at  a  lower  temperature  than  the  second. 


213]  THERMODYNAMICS  263 

There  are  many  most  important  consequences  of  this  prin- 
ciple ;  but  they  are  not  suited  for  presentation  here. 

A  heat-engine  is  a  mechanism  which  allows  some  "  work- 
ing-substance," like  steam,  to  receive  heat-energy  at  a  high 
temperature ;  allows  the  working-substance  to  do  external 
work,  in  doing  which  its  temperature  falls ;  then  by  means 
of  external  work  brings  the  working-substance  back  to  its 
original  condition ;  etc.  indefinitely.  While  the  working- 
substance  is  being  brought  back  to  its  original  condition,  it 
must  give  out  some  heat-energy  to  some  external  body. 
Let  hi  be  the  heat-energy  received,  7i2  be  the  heat-energy 
given  out.  Then,  by  the  conservation  of  energy,  the  ex- 
ternal work  done  by  the  substance  in  excess  of  that  done 

on  it  is 

W=hl-k2 (1) 

because  the  working-substance  is  back  in  its  original 
condition ;  and  so  no  internal  work  has  been  done,  only 
external. 

The  "  efficiency "  of  a  heat-engine  is,  by  definition,  the 
ratio  W/hi;  and  it  may  be  proved  by  the  second  prin- 
ciple of  thermodynamics  that  this  efficiency  cannot,  un- 
der definite  conditions  of  temperature,  exceed  a  certain 
limit.  An  engine  which  has  the  greatest  possible  effi- 
ciency is  called  a  perfect  engine,  and  it  may  be  proved 
that  the  efficiency  of  such  an  engine  is  (except  for  a 
small  number  of  bodies)  entirely  independent  of  the 
working-substance,  being  the  same  for  all  fluids  and  also 
certain  solids.  But  the  efficiency  does  depend  upon  the 
conditions  of  temperature.  If  the  body  from  which  the 
heat-energy  is  taken  into  the  working-substance  has  an 
absolute  temperature,  Ti  (t°  C.  +  273),  and  if  the  abso- 
lute temperature  of  the  coldest  large  body  available 
for  the  working-substance  to  give  up  heat-energy  to  is 
T2,  it  may  be  proved  that  the  efficiency  of  a  perfect  gas- 

m   rjp 

engine  is  — ^ — - .     But  the  efficiency  of  any  other  perfect 


264  THEORY   OF  PHYSICS  [CH.  VII 

engine  working  between  the  same  temperatures  must  be 
the  same.  (This  leads  to  a  numerical  determination  of 
temperature,  which  is  entirely  independent  of  the  nature  of 
the  thermometer.) 

The  greatest  possible  efficiency  is,  of  course,  unity  ;  i.  e. 
under  these  conditions, 


_ 


Hence  Tz  must  equal  0.  A  lower  temperature  than  this 
would  mean  an  efficiency  greater  than  one,  which  is  im- 
possible. So  the  lowest  possible  temperature  is  that  for 
which  T=0  or  -273°  centigrade.  This  is  called  the 
"absolute  zero." 


BOOK  IV 

ELECTEICITY  AND   MAGNETISM 


BOOK    IV 

ELECTRICITY    AND    MAGNETISM 


INTRODUCTION 

IT  has  been  known  for  many  hundred,  if  not  thousand, 
years  that,  when  a  piece  of  amber  is  rubbed  with  flannel, 
it  has  the  power  of  "  attracting  "  small  portions  of  matter, 
and  that  a  magnetic  needle  suspended  on  a  pivot  turns  so 
as  to  point  in  a  northerly  direction.  Again,  no  phenom- 
enon has  been  observed  with  such  unfailing  interest  as 
the  discharge  of  lightning.  These  phenomena  are  all  re- 
lated to  each  other;  and  the  study  of  their  laws  and  those 
of  other  phenomena  of  the  same  nature  forms  the  subject 
of  "  Electricity  and  Magnetism." 

214.  All  electric  and  magnetic  effects  are  intimately 
associated  with  the  "  ether,"  that  medium  which  conveys 
the  waves  discussed  in  Eadiation,  Chapter  V.  of  HEA/T  ; 
and  it  may  be  well  to  describe  briefly  its  properties.  It  is 
a  medium  which  penetrates  all  the  spaces,  large  and  small, 
between  the  portions  of  ordinary  matter ;  in  fact,  the  ether 
may  be  regarded  as  a  universal  medium  in  which  the  mi-, 
nute  portions  of  ordinary  matter  are  immersed.  The  ether 
has  inertia,  as  is  proved  by  the  fact  that  waves  in  it  travel 
with  a  finite  velocity  ;  but  its  other  properties  are  uncertain. 
If  it  is  made  up  of  parts,  as  ordinary  matter  is,  they  must 
be  extremely  minute ;  because  waves  in  any  medium  are 
always  immensely  longer  than  the  dimensions  of  the  ulti- 


268  THEORY  OF  PHYSICS 

mate  particles  of  that  medium,  and  ether-waves  are  them- 
selves very  short.  Ether-waves  are,  however,  transverse 
(as  will  be  shown  later) ;  and  this  means  that  the  ether  has 
rigidity,  or  resists  a  change  in  shape  (see  Art.  74).  So  the 
ether  may  have  a  strain  of  this  type ;  and,  if  it  does,  it  will 
have  potential  energy.  The  ether  can  also  move  ;  and  then 
it  will  have  kinetic  energy.  If  there  is  any  ordinary  mat- 
ter immersed  in  the  -ether,  it  is  connected  to  the  ether  in 
some  way ;  so  that  all  the  properties  of  the  ether  are 
affected  by  the  presence  of  the  matter,  it  is  "  loaded  down  " 
with  it.  If  the  ether  is  strained,  so  also  are  any  portions 
of  matter  in  it ;  and  the  apparent  inertia  of  the  ether  is 
evidently  increased  by  the  presence  of  ordinary  matter. 
Consequently,  in  all  electric  and  magnetic  phenomena  the 
effect  of  matter  is  most  important.  It  would  be  expected 
that  different  kinds  of  matter  would  produce  different 
effects ;  and  that,  in  the  case  of  waves,  those  of  different 
length  would  not  be  affected  alike.  Such  is  observed  to 
be  the  case. 


CHAPTEK  I 

GENERAL  PROPERTIES  OF  ELECTRIC  CHARGES 

215.  Production  of  Charges.  If  two  different  bodies,  e.  g. 
glass  and  silk,  or  sealing-wax  and  flannel,  are  rubbed  to- 
gether so  as  to  bring  their  surfaces  into  intimate  contact, 
and  are  then  separated,  two  things  may  be  observed : 

(1)  Work  is  required  to  separate  them  ;   and  so  potential 
energy  is  stored  up  somewhere,  in  the  form  of  a  strain. 
This  is  proved  by  the  fact  that,  if,  after  separation  to  a 
short  distance,  the  two  bodies  are  set  free  to  move,  they 
will  move  towards  each  other.     (Compare  raising  a  body 
off   the    earth,   the    stretching    of   an    elastic    cord,    etc.) 

(2)  Each  body  now  has  the  power  of  causing  small  bits  of 
matter,  like  dust,  pieces  of  paper,  pieces  of  gold-foil,  etc., 
to  move  towards  it.     (This  property  will  be  shown  in 
Article  221  to  be  due  to  changes  in  the  strain  of  the  me- 
dium around  the  charged  body.) 

This  last  property  is  generally  taken  as  the  test  of 
"  electrification."  A  body  which  has  it  is  said  to  be  "  elec- 
trified "  or  "  charged,"  or  to  have  a  "  charge  "  of  electricity. 
If  two  bodies  which  differ  in  the  least  in  the  nature  or 
arrangement  of  their  molecules  are  brought  into  contact  in 
any  way,  e.  g.  by  pressure  or  by  friction,  they  are  found, 
on  separation,  to  be  charged. 

There  are  many  other  methods  of  producing  charges; 
and  some  will  be  discussed  in  later  chapters.  One  of  the 
commonest  is  simply  to  touch  a  charged  body  with  an 
uncharged  one,  when  some  of  the  charge  passes  to  the 
latter. 


270  THEORY  OF  PHYSICS  [CH.  I 

The  fact  that  the  medium  around  a  charged  body  is 
strained  is  proved  also  by  the  disruption  of  air,  glass,  etc. 
by  ordinary  sparks  from  electric  machines.  The  strain 
primarily  is  in  the  ether,  as  is  proved  by  the  possibility 
of  having  charged  bodies  in  a  vacuum ;  but,  if  ordinary 
matter  is  present,  it  as  well  as  the  ether  is  the  seat  of  the 
strain  (except  in  conductors.  See  Art.  220).  It  will  be- 
come evident  in  the  following  paragraphs  that  the  essential 
feature  of  electrification  is  the  particular  strain  which  is 
characteristic  of  it. 

216.  Positive  and  Negative  Charges.  Although  all  charges 
can  produce  motions  in  small  portions  of  matter,  there  are 
different  kinds  of  charges.  This  may  be  shown  by  the 
following  experiments :  Suspend  in  a  paper  stirrup  carried 
by  a  long  thread  a  glass  rod  which  has  been  rubbed  with 
silk ;  bring  near  its  end  another  glass  rod  which  has  also 
been  rubbed  with  silk  ;  there  is  repulsion.  If  a  piece  of 
sealing-wax  (or  ebonite)  which  has  been  rubbed  with  flan- 
nel is  brought  near  the  suspended  glass  rod,  there  is  attrac- 
tion. If  the  charged  sealing-wax  is  suspended  in  the 
stirrup  instead  of  the  charged  glass  rod,  it  may  be  observed 
that  a  glass  rod  charged  by  rubbing  with  silk  will  produce 
attraction,  while  another  rod  of  sealing-wax  charged  by 
rubbing  with  flannel  will  produce  repulsion. 

Thus,  two  identical  bodies  charged  by  the  same  process 
cause  repulsion.  It  is  also  found,  by  experiments  per- 
formed on  the  suspended  glass  rod  and  the  suspended  seal- 
ing-wax, that  all  other  charged  bodies  may  be  divided  into 
two  classes,  —  some  attract  the  charged  glass,  others  repel 
it ;  but  those  which  attract  the  glass  repel  the  charged  seal- 
ing-wax, and  those  which  repel  the  glass  attract  the  sealing- 
wax.  Glass  rubbed  with  silk  belongs  to  the  second  class  ; 
sealing-wax  rubbed  with  flannel,  to  the  first.  And  the 
general  law  may  be  stated,  that  "  all  charges  of  the  same 
class  repel  each  other,  and  attract  charges  of  the  other 
class." 


217]     GENERAL  PROPERTIES  OF  ELECTRIC  CHARGES     271 

The  same  material  may  belong  to  both  classes,  depend- 
ing upon  what  other  body  it  has  been  in  contact  with. 
Thus,  glass  rubbed  with  the  fur  of  a  cat's  skin  attracts 
glass  rubbed  with  silk.  So  the  above  division  is  one  of 
charges,  not  of  materials. 

These  two  classes  of  charges  have  received  the  names 
"  positive  "  and  "  negative."  One  reason  for  this  choice  is 
that,  if  one  kind  of  a  charge  produces  any  kind  of  motion 
in  a  charged  body,  the  other  kind  of  a  charge  will  produce 
exactly  the  opposite  motion  of  the  same  body ;  just  as  a 
positive  force  or  moment  produces  exactly  opposite  effects 
to  those  of  a  negative  force  or  moment. 

By  common  consent,  the  kind  of  charge  which  a  glass 
rod  takes  when  rubbed  with  silk  is  called  "  positive ; "  and 
so  the  nature  of  any  charge  may  be  at  once  determined 
by  comparing  it  with  that  on  the  glass. 

217.  "  Specific  Attraction  "  of  Matter  for  Electricity.  When 
two  different  bodies  are  brought  into  contact,  and  then 
separated,  attraction  is  observed  between  the  two  ;  that  is, 
one  is  charged  positively,  the  other  negatively.  The  ques- 
tion as  to  which  will  be  positive  and  which  negative  de- 
pends upon  the  properties  of  the  two  bodies  with  relation 
to  each  other. 

When  any  one  body  becomes  charged,  there  is  a  change 
in  the  potential  energy  of  the  molecules  of  the  body ;  but 
this  change  may  be  different  for  a  positive  charge  and  for 
a  negative  one.  Therefore,  since  it  is  a  general  law  of 
nature  that  potential  energy  tends  to  get  as  small  as  pos- 
sible, this  particular  body  will  have  a  tendency  to  become 
charged  with  that  charge,  +  or  — ,  which  will  produce  the 
greater  decrease  in  the  potential  energy.  This  is  some- 
times expressed  by  saying  that  each  kind  of  matter  has  a 
"  specific  attraction  "  of  a  definite  amount  for  either  posi- 
tive or  negative  electricity. 

If,  then,  two  bodies  are  brought  in  contact  and  charges 
are  produced,  that  one  which  has  the  greater  specific  at- 


272  THEORY  OF  PHYSICS  [CH.  I 

traction  for  a  positive  charge  will  become  positively  charged, 
and  the  other  negatively.     (See  the  following  article.) 
I      It  is  possible,  of  course,  to  arrange  a  table  of  substances 
in  which  they  will  stand  in  the  order  of  their  specific  at- 
tractions for  positive  charges. 

218.  Quantities  of  Electricity.  Since  charges  produce  cer- 
tain motions  (or  "  forces ")  in  other  charged  bodies,  the 
equality  of  two  charges  may  be  tested  by  seeing  if  they 
will  produce  identically  the  same  effect  on  a  third  charged 
body  under  the  same  conditions.  If  they  do,  they  are  said 
to  have  equal  "  quantities "  of  electricities.  If,  however, 
one  charge  produces  a  force  which  is  exactly  the  opposite 
of  that  produced  by  the  second  one,  the  two  charges  have 
equal  quantities,  but  are  of  opposite  kinds.  One  has  a 
charge  +  0 ;  the  other,  a  charge  —  e. 

If  two  bodies  are  brought  in  contact  and  separated,  it 
may  be  observed  that  one  is  charged  positively,  and  the 
other  negatively ;  and  it  may  be  proved  by  most  careful 
experiments  that  the  quantity  of  positive  charge  of  the 
one  exactly  equals  that  of  the  negative  charge  of  the  other. 
In  other  words,  a  positive  charge  can  never  be  obtained 
without  at  the  same  time  there  being  an  equal  negative 
charge  produced.  This  may  be  regarded  as  a  necessary 
consequence  of  the  fact  .that  the  medium  around  charges  is 
strained.  For  a  'strain  can  be  maintained  only  by  two 
equal  and  opposite  forces  ;  e.  g.  a  wire  stretched  or  twisted, 
a  liquid  compressed,  etc.  So  in  an  electric  strain  there 
must  be  equal  and  opposite  phenomena  at  the  ends  of  the 
strain  ;  that  is,  where  it  starts  and  ends. 

In  order  to  measure  quantities  of  electricity,  that  is,  in 
order  to  give  a  numerical  value  to  a  charge,  it  is  necessary 
to  adopt  some  unit  charge  to  serve  as  a  standard.  The 
unit  adopted  for  certain  measurements  is  defined  in  this 
way :  two  unit  charges  placed  at  a  distance  of  one  centi- 
metre apart  in  air  have  a  force  between  them  of  one  dyne. 
If  one  charge  is  positive  and  the  other  negative,  the  force 


219]  GENERAL  PROPERTIES  OF  ELECTRIC  CHARGES   273 

will  be  one  of  attraction  ;  otherwise  there  will  be  repulsion. 
This  force  can,  of  course,  be  measured  (at  least  theoreti- 
cally) by  means  of  a  spiral  spring  or  a  chemical  balance. 
The  unit  thus  defined  is  called  the  "  Electrostatic  Unit  of 
Quantity,"  because  it  is  used  in  measuring  charges  which 
are  at  rest. 

219.  Law  of  Electrostatic  Force.  The  force  between  two 
charged  bodies  (i.  e.  the  change  in  momentum  in  one 
second  which  would  be  produced  in  each  body  if  free  to 
move)  is  found  to  depend  upon  three  things  :  the  quanti- 
ties of  the  charges,  their  distance  apart,  and  the  material 
medium  in  which  the  charged  bodies  are  placed,  e.  g.  air, 
water,  paraffin,  etc.  The  exact  law  of  the  action  may  be 
expressed, 


where  e  and  ef  are  the  quantities  of  the  two  charges  ;  r  is 
their  distance  apart;  and  K  is  a  constant  for  any  one 
medium,  but  different  for  different  media,  F  is  the  me- 
chanical force,  and  is  measured  in  dynes  ;  it  is  positive 
for  a  repulsion,  because,  if  e  and  e'  both  have  the  same  sign, 
i.  e.  if  there  is  repulsion,  F  is  evidently  positive. 

On  the  electrostatic  system  of  units,  K  has  the  value  1 
for  air,  for  a  unit  charge  is  so  defined  that,  if  placed  in 
air  at  a  distance  of  1  cm.  from  an  equal  charge,  there  is 
a  force  of  1  dyne.  That  is,  if  e  =  er  =  1  and  r  =  1  in  air,  F—  1. 
Hence  K=  1  for  air.  For  other  media  K  has  a  value  always 
(except  in  certain  gases  at  low  pressure)  greater  than  for 
air.  On  any  other  system  of  units,  than  the  electrostatic 
one,  K  would  not  equal  1  for  air.  K  varies  with  the  tem- 
perature, and  also  the  pressure,  if  the  medium  is  a  gas. 

This  law  of  force  is  called  "  Coulomb's  Law,"  because  he 
first  verified  it  experimentally.  His  method  was  the  direct 
one  of  placing  charged  bodies  at  certain  distances  apart 
and  comparing  the  forces.  A  better  method  is  to  make 


274  THEORY  OF  PHYSICS  [CH.  I 

deductions  from  the  law,  and  see  if  they  are  all  ful- 
filled. Such  has  been  done,  and  it  may  be  proved  that 
many  phenomena  are  easily  explained  as  consequences  of 
this  law,  and  cannot  be  accounted  for  by  any  other  law. 

220.  Conductors  and  Dielectrics,  It  is  easily  proved  by 
experiment  that,  so  far  as  electric  charges  are  concerned, 
there  are  two  classes  of  material  bodies.  If  a  charge  is 
placed  on  a  body  of  one  class,  it  stays  at  the  point  where  it 
is  placed ;  e.  g.  if  a  certain  portion  of  a  glass  rod  is  rubbed 
with  silk,  that  particular  portion  is  the  only  part  of  the 
glass  which  receives  a  charge.  Such  bodies  are  called  "  in- 
sulators" or  "non-conductors." 

But,  if  a  charge  is  placed  on  a  body  of  the  other  class,  it 
spreads  and  appears  over  its  entire  outer  surface.  Such  a 
body  is  called  a  "  conductor ; "  and  all  metals  are  illustra- 
tions. The  fact  that  the  charge  on  a  conductor  is  entirely 
on  the  outer  surface  is  most  important.  It  can  be  proved 
by  delicate  experiments.  If  the  conductor  is  hollow,  the 
entire  charge  is  on  the  outer  surface,  as  stated,  unless 
another  charged  body  is  placed  inside,  in  the  hollow  space, 
in  which  case  there  will  be  a  charge  on  the  inner  surface 
of  the  conductor  caused  by  the  presence,  in  the  interior,  of 
the  second  charged  body.  (See  Art.  232.) 

If  two  conductors '  are  charged,  it  is  possible  to  keep  the 
charges  unchanged,  only  if  the  two  conductors  are  sepa- 
rated by  means  of  an  insulating  substance.  For,  if  a  con- 
ductor is  used,  the  charges  spread  over  it,  and  so  everything 
is  changed.  If  the  original  charges  were  e  and  e',  the 
entire  charge  when  the  two  conductors  are  joined  by  a 
third  one  is  found  by  experiment  to  be  in  every  case  e  +  e'. 
So,  if  e'  =  —  e,  that  is,  if  the  two  charges  were  equal  and 
opposite,  the  final  charge  is  zero,  the  two  charges  have  neu- 
tralized each  other.  Any  uncharged  conductor  may  there- 
fore be  considered  as  having  on  its  surface  two  equal  and 
opposite  charges  of  any  amount  desired,  and  it  is  possible, 
as  will  be  shown  later  (Art.  229),  to  separate  these  charges. 


220]     GENEKAL  PROPERTIES  OF  ELECTRIC  CHARGES      275 

The  following  table  contains  the  names  of  some  non- 
conductors and  conductors  :  — 

TABLE 

Non-Conductors.  Conductors. 

Glass.  All  metals. 

Paraffin.  Salt  water. 

Dry  air.  .        Moist  cotton. 

Ebonite.  The  human  body. 

Silk.  Damp  wood. 

Porcelain. 

Shellac. 

Wool. 

Resin. 

Oils. 

The  facts  that  a  charge  spreads  itself  over  the  surface 
of  a  conductor,  and  that  consequently,  in  order  to  keep 
a  charge  on  a  conductor,  it  must  be  surrounded  by  a  non- 
conductor, may  be  explained,  if  it  is  remembered  that  the 
essential  feature  of  a  charge  is  the  strain  in  the  surrounding 
medium.  If  glass  and  silk  are  rubbed  together  and  then 
separated,  the  medium  in  between  them  and  around  them 
is  strained,  as  already  explained.  The  electric  strain  in 
the  medium  begins  and  ends  on  the  charges,  just  as,  in  the 
mechanical  strain  produced  in  a  wire  by  stretching  it,  the 
strain  is  bounded  by  the  two  ends  where  the  equal  and 
opposite  forces  are  applied.  In  fact,  the  electric  charges 
may  best  be  regarded  as  simply  the  phenomena  at  the  ends 
of  the  strain.  To  be  stained,  the  medium  must  have  some 
elasticity ;  but  imagine  a  medium  which  has  none,  which 
offers  no  opposition  to  any  attempt  to  strain  it.  It  would 
be  impossible  to  keep  a  positive  and  negative  charge  apart 
by  such  a  medium.  Further,  imagine  a  charge  placed  at 
any  point  in  the  air ;  the  air  will  be  strained  on  all  sides 
of  the  charge.  Now  place  this  charge  on  a  small  portion 
of  a  medium  like  the  one  described,  which  offers  no  resist- 


276 


THEORY  OF  PHYSICS 


[CH.  I 


ance  to  be  strained ;  there  can  be  no  strain,  of  course,  in 
this  medium,  and  so  the  strain  in  the  surrounding  air 
must  begin  at  the  outer  surface  of  the  inner  medium. 
Consequently,  the  action  is  just  as  if  the  charge  had  been 
spread  over  the  surface  of  that  medium.  So,  a  conductor 
behaves  exactly  like  a  medium  which  cannot  resist  any 
attempt  to  strain  it,  which  yields  at  once  to  the  slightest 
electric  stress.  A  non-conductor,  on  the  other  hand,  can 
be  electrically  strained  ;  and,  since  the  essential  features  of 
charges  are  in  the  strained  medium,  a  special  name  has 
been  given  non-conductors  so  as  to  emphasize  this  fact : 
they  are  called  "  dielectrics." 

(eef  \ 
F  =  ^-2  J  the  quantity  K 


is  a  constant  for  any  one  medium,  and  so  it  is  called  the 
"  dielectric  constant."  Since  in  all  conducting  media  a 
strain  is  impossible,  K  must  for  them  be  infinitely  great. 
Its  value  for  various  dielectrics  will  be  given  later. 

Electroscopes.  The  fact  that  two  bodies 
which  are  similarly  charged  repel  each 
other  may  be  made  use  of  in  an  instru- 
ment designed  to  detect  charges.  One 
form  of  instrument  is  called  the  "  gold- 
leaf  electroscope."  It  consists  of  a  metal 
plate  to  which  is  attached  a  metal  rod 
carrying  at  its  lower  end  two  long  nar- 
row strips  of  gold-foil.  These  are  sus- 
pended, as  shown,  in  a  glass  jar,  so  as  to 
protect  them  from  disturbances.  As  just 
explained,  a  metal  body  of  any  shape  has 
the  property  of  allowing  a  charge  placed 
FIG.  146.  a^  one  point  of  it  to  spread  over  the  en- 

tire surface.  So,  if  a  charge  of  any  kind 
is  given  the  top  metal  plate^  it  will  spread  over  the  gold- 
leaves  ;  and,  as  they  are  similarly  charged,  they  will  repel 
each  other  and  stand  diverged.  If,  now,  another  similar 


221]  GENERAL  PROPERTIES  OF  ELECTRIC  CHARGES  277 

charge  is  added,  the  leaves  will  diverge  still  farther ;  while, 
if  the  charge  is  of  the  opposite  kind,  it  will  neutralize 
some  of  the  existing  charge,  and  the  leaves  will  come  closer 
together. 

As  will  be  explained  later,  in  speaking  of  induction 
(Art.  230),  it  is  not  necessary  actually  to  add  the  charge 
to  the  metal  plate  in  order  to  produce  the  motions  of  the 
leaves.  If  the  charged  body  is  simply  brought  near,  the 
leaves  will  diverge  more  or  will  collapse,  depending  upon 
the  nature  of  the  charges. 

Frictional  Machines.  The  general  method  of  producing  a 
charge  is  to  rub  together  two  different  dielectrics,  e.  g. 
glass  and  silk,  and  then  to  separate  them.  If  the  charges 
are  removed  from  the  two  bodies,  fresh  charges  may  be 
produced  by  repeating  the  process,  etc.  This  succession  of 
steps,  rubbing,  separation,  discharge,  may  be  made  auto- 
matic, as  it  is  in  the  so-called  "  frictional  machines."  The 
charges  are  removed  from  the  charged  dielectrics  by  pass- 
ing conductors  over  them,  or  by  allowing  conductors  pro- 
vided with  sharp  points  to  pass  near  them.  As  will  be 
explained  later  (see  Art.  230),  this  last  method  also  pro- 
duces the  discharge  of  the  charged  dielectric  and  the 
appearance  of  an  equal  charge  on  the  conductor. 

221.  Energy  of  Medium.  As  stated  before,  when  two 
bodies  are  charged  by  contact  and  separation,  the  energy 
which  is  associated  with  the  charges  is  in  the  surrounding 
medium ;  and  this  space  is  sometimes  called  the  "  electric 
field."  The  amount  of  energy  is  equal  to-  the  work  done 
in  separating  the  charges,  and  some  idea  may  be  formed  of 
relative  amounts  from  a  consideration  of  the  formula 

eef 
F  =  — y .     It  is  evident  that  the  greater  the  charges,  so 

much  the  more  work  must  be  done  to  separate  them.  Fur- 
ther, K  is  different  for  different  media,  being  greater  for 
almost  all  substances  than  for  air ;  so  more  work  is  required 
to  separate  the  charges  in  air  than  in  other  dielectrics. 


278  THEORY  OF  PHYSICS  [CH.  I 

Consequently,  there  is  more  energy  per  cubic  centimetre  in 
air  than  in  other  dielectrics  under  similar  conditions.  The 
amount  of  the  energy  in  one  cubic  centimetre  must  also  be 
greater  near  the  charges  than  farther  away,  because  r  is 
small  there.  Since  there  is  no  strain  at  all  inside  a  con- 
ductor, there  is  no  energy  inside ;  and  so,  as  far  as  electro- 
.static  energy  is  concerned,  a  conductor  acts  like  a  medium, 
for  which  K  is  infinitely  great,  as  was  noted  above. 

Motions  in  Electric  Fields.  Since  all  motions  which  take 
place  of  themselves  do  so  in  such  a  way  as  to  cause  a  de- 
crease of  potential  energy,  it  must  be  possible  to  "  explain  " 
thus  all  motions  in  electric  fields. 

Two  bodies,  one  charged  positively,  the  other  negatively, 
attract  each  other,  because  work  is  required  to  separate 
them;  and  so,  by  coming  closer  together,  the  potential 
energy  becomes  less.  (Compare  a  body  falling  to  the 
earth.)  Two  bodies  charged  the  same  way  repel  each 
other ;  because,  by  so  moving  apart,  the  strain  in  the  field 
becomes  less. 

If  a  piece  of  dielectric  for  which  K  is  greater  than  for 
air  (e.  g.  a  bit  of  glass,  paper,  paraffin)  is  placed  near  a 
charged  body  in  air,  the  immediate  effect  is  to  weaken  the 
strain  and  so  to  decrease  the  energy  in  that  portion  of  space 
occupied  by  the  dielectric  which  is  inserted.  But,  if  this 
same  piece  of  dielectric  is  placed  nearer  the  charged  body, 
where  the  strain  would  naturally  be  greatest,  the  decrease  in 
the  energy  is  greater  than  it  would  be  in  a  position  farther 
away.  Consequently,  since  by  the  piece  of  dielectric  mov- 
ing nearer  to  the  charged  body  the  potential  energy  is 
decreased,  the  motion  will  take  place  of  itself.  In  other 
words,  a  charged  conductor,  either  +  or  — ,  will  attract  such 
a  piece  of  dielectric  in  air.  Or,  in  general,  such  a  piece  of 
dielectric,  if  placed  in  air,  will  tend  to  move  from  a  place 
where  the  force  is  weak  to  one  where  it  is  strong.  Of 
course,  if  JTfor  the  dielectric  was  less  than  for  air,  the  mo- 
tion would  be  in  the  opposite  direction.  By  experiments 


222]     GENERAL  PROPERTIES  OF  ELECTRIC  CHARGES     279 

of  this  nature,  the  value  of  K,  compared  with  its  value  for 
air,  may  be  found  for  various  dielectrics. 

Since  K  for  a  conductor  may  be  regarded  as  infinite,  a 
piece  of  a  conducting  substance  will  be  attracted  by  a 
charged  body  much  more  than  any  piece  of  dielectric.  (Of 
course,  to  produce  motion  actually,  the  piece  of  matter  to 
be  attracted  must  be  suspended  perfectly  free  to  move,  or 
else  must  be  very  light  in  weight.) 

222.  "  Lines  of  Force."  An  "electric  field"  has  been  de- 
fined as  the  name  given  to  the  space  surrounding  charged 
bodies,  in  which  electric  forces  may  be  perceived.  The 
properties  of  the  field  may  be  best  represented  by  drawing 
certain  lines  in  it  which  mark  the  direction  of  the  force  at 
each  point  in  the  field.  A  minute  positively  charged  body, 
placed  at  any  point  in  the  field,  will  tend  to  move  in  some 
direction.  If  the  body  is  so  small  as  to  be  considered  a 
"particle"  (see  Art.  27),  this  direction  is  called  the  direc- 
tion of  the  force  at  that  point  where  it  is  placed.  A  "  line 
of  force  "  is  such  a  line  that  at  each  of  its  points  its  direc- 
tion is  that  of  the  force  at  that  point.  (A  charged  particle, 
if  left  free  to  move,  would  not  continue  to  move  along  a 
line  of  force;  owing  to  the  inertia  of  the  matter  com- 
posing the  particle,  it  would  at  any  instant  have  a  certain 
momentum,  and  the  actual  direction  of  its  motion  would 
be  determined  by  the  geometrical  sum  of  this  mqnientum 
and  that  produced  by  the  electric  force.)  It  should  be 
particularly  noticed  that  lines  of  force  cannot  be  drawn 
in  conductors,  because  in  a  conductor  there  is  no  force  if 
the  charges  are  not  changing,  as  already  explained.  So 
lines  of  force  can  be  drawn  only  in  dielectrics.  But,  fur- 
ther, inside  a  hollow  closed  conductor  no  lines  of  force  can 
be  drawn ;  for  a  line  of  force  must  start  from  a  positive 
charge,  and,  as  explained  before,  there  are  no  charges  inside 
a  closed  conductor. 

Thus,  lines  of  force  always  start  from  positively  charged 
bodies,  and  end  on  negatively  charged  ones.  If  the  field  of 


280 


THEORY  OF  PHYSICS 


[CH.  I 


force  is  due  to  a  positively  charged  spherical  conductor 
(the  equal  negative  charge  being  removed  to  an  infinite 
distance),  the  lines  of  force  are  straight  lines  drawn  per- 
pendicular to  the  surface  in  all  directions. 

Other  illustrations  of  lines  of  force  are  given  in  the 
accompanying  diagrams ;  they  are  taken,  almost  without 
change,  from  J.  J.  Thomson's  "  Elements  of  Electricity  and 
Magnetism,"  and  the  lines  are  so  drawn  as  to  represent  by 
their  numbers  the  relative  charges  on  the  different  bodies. 
The  first  represents  the  lines  of  force  due  to  two  equal  and 
opposite  charges. 


FIG.  147. 

The  second  represents  the  lines  of  force  due  to  two  equal 
charges,  either  positive  or  negative.  If  they  are  positive, 
the  lines  are  leaving  the  bodies  ;  if  they  are  negative,  they 
are  ending  on  them.  (The  other  ends  of  the  lines  are  on 
the  equal  opposite  charges  which  are  supposed  to  be  off  at 
an  infinite  distance.) 


FIG.  148. 


222]     GENERAL  PROPERTIES  OF  ELECTRIC  CHARGES       281 

The  third  represents  the  lines  of  force  due  to  a  positive 
charge  four  times  as  great  as  the  negative  charge. 


FIG.  149. 

The  fourth  represents  the  lines  of  force  due  to  a  charge 
at  A,  four  times  as  great  as  a  similar  charge  at  B. 


FIG.  150. 

» 

The  fifth  represents  the  lines  of  force  due  to  a  charge  on 
a  conductor  formed  of  two  spherical  surfaces  cutting  at  right 
angles. 


FIG.  161. 


282 


THEORY   OF  PHYSICS 


[CH.  I 


The  sixth  represents  the  lines  of  force  due  to  two  equal 
and  opposite  charges  placed  on  two  parallel  conducting 
planes. 


\ 


FIG.  152. 

In  this  last  it  is  evident  that  the  strain  is  almost  entirely 
in  the  space  between  the  parallel  plates ;  the  charges  have 
arranged  themselves  as  they  have,  so  as  to  make  the  po- 
tential energy  as  small  as  possible.  In  this  space,  too, 
the  lines  of  force  are  parallel  at  points  away  from  the 
edges  of  the  plates;  and  such  a  field  of  force  is  called 
"  uniform." 

By  studying  these  diagrams,  it  is  seen  that,  where  the 
lines  of  force  are  most  numerous,  there  the  electric  force  is 
strongest ;  and  a  great  deal  may  be  learned  about  the  prop- 
erties of  any  field  by  a  consideration  of  the  lines  of  force. 
One  thing  is  evident :  as  two  oppositely  charged  bodies 
approach  each  other,  the  lines  of  force  become  shorter ;  and 
so  it  is  sometimes  said  that  "  lines  of  force  tend  to  con- 
tract ; "  but  this  is,  of  course,  merely  a  description. 


CHAPTEK  II 
ELECTRIC  POTENTIAL  AND  INDUCTION 

IN  order  to  discuss  more  fully  the  properties  of  electric 
charges,  it  is  necessary  to  introduce  some  mathematical 
ideas  and  definitions.  As  was  seen  in  Chapter  I.,  the  most 
important  laws  of  charged  bodies  depend  upon  relative 
amounts  of  potential  energy ;  and  the  same  statement  is 
true  of  the  charges  themselves. 

223.  Electric  Potential.  The  amount  of  external  work 
necessary  to  move  a  unit  positive  charge  from  a  point  P  in 


FIG.  153. 

the  field  to  a  point  Q  is  called  the  "difference  of  potential" 
between  Q  and  P.  If  this  difference  of  potential  is  repre- 
sented by  E,  the  work  done  in  making  a  positive  charge, 
e,  pass  from  P  to  Q  is  eE.  As  a  result  of  this  work, 
an  equal  amount  of  potential  energy  is  given  the  sur- 
rounding dielectric.  E  may  be  a  positive  or  a  negative 
quantity.  If  it  is  positive,  work  is  actually  done  by  some 
external  cause,  and  energy  is  added  to  the  medium.  If  it 
is  negative,  the  medium  loses  energy,  and  the  change  takes 
place  of  itself,  the  energy  being  spent  in  giving  kinetic 
energy  to  the  body  which  has  the  charge,  or  in  doing  some 
external  work.  Thus,  if  there  is  a  positive  charge  in  the 


284  THEORY  OF  PHYSICS  [CH.  II 

field  at  A,  and  if  Q  is  nearer  A  than  P  is,  the  difference 
of  potential  E  is  positive,  because  work  must  be  done  by 
external  forces  to  make  a  positive  charge  pass  from  P  to  Q. 
If  there  is  a  negative  charge,  though,  at  A,  the  difference 
of  potential  E  is  negative,  because  a  positive  charge  would 
move  of  itself  from  P  to  Q. 

Another  definition  of  the  difference  of  potential  E  is 
that  it  is  the  amount  of  work  which  a  unit  positive  charge 
will  do  if  it  moves  of  itself  from  Q  to  P.  Still  another  is 
that  E  is  the  amount  of  work  required  to  carry  a  unit 
negative  charge  from  Q  to  P. 

These  amounts  of  work,  or  changes  in  potential  energy, 
are  entirely  independent  of  the  path  followed  in  moving 
between  P  and  Q.  (If  it  were  not  so,  it  would  be  possible 
to  get  "  perpetual  motion.") 

224.  Electric  Forces.  Sparks.  If  E  is  positive,  external 
work  is  required  if  a  unit  positive  charge  is  carried  from  P 
to  Q ;  and  so  a  force  must  have  been  overcome.  The  direc- 
tion of  the  force  at  any  point  has  been  defined  as  the  direc- 
tion in  which  a  positive  charge  would  move  as  a  result  of 
the  force,  if  placed  at  that  point ;  and  so  the  direction  of  the 
force  in  this  case  between  P  and  Q  is  from  Q  towards  P, 
since  E  is  positive.  If  E  is  negative,  the  direction  of  the 
force  is  from  P  to  Q,  because  a  positive  charge  would  move 
of  itself  in  that  direction.  If  E  =  0,  i.  e.  if  there  is  no  dif- 
ference of  potential,  no  work  is  required  to  pass  from  P  to 
Q,  or  vice  versa.  This  implies  that  there  is  no  force  oppos- 
ing or  helping  the  motion.  If  there  is  a  force,  there  must 
be  a  difference  of  potential. 

If  the  difference  of  potential  between  two  points  is  great, 
it  is  equivalent  to  saying  that  the  electric  force  is  great 
between  the  two.  Consequently,  the  strain  in  the  dielec- 
tric of  the  field  must  be  great.  This  condition  of  affairs 
may  be  secured  by  placing  near  each  other  two  bodies,  one 
charged  positively,  the  other  negatively.  There  is  now  a 
great  difference  of  potential  between  two  points,  one  on 


225]  ELECTRIC  POTENTIAL  AND  INDUCTION  285 

each  charged  body ;  and  there  is  a  great  strain  in  the  inter- 
vening dielectric.  If  the  strain  is  too  great,  there  is  a  rup- 
ture of  the  dielectric,  a  spark  passes,  and  the  two  charges 
combine.  A  spark  acts  like  a  perfect  conductor  connect- 
ing the  two  bodies ;  it  consists  of  gases  which  are  made 
luminous  by  the  rupture  of  the  medium  and  the  motion  of 
the  charges. 

A  spark  will,  then,  pass,  in  general,  at  points  where  the 
strain  is  greatest ;  that  is,  where  the  difference  of  potential 
is  greatest  or  the  electric  field  most  intense.  If  the  die- 
lectric is  weakened  at  any  point,  a  spark  will,  of  course, 
pass  there  more  easily. 

225.  Potential  at  a  Point.  As  will  be  immediately  proved 
in  the  following  article,  there  can  be  no  difference  of  poten- 
tial between  any  two  points  of  a  conductor  on  which  the 
charges  are  at  rest.  (If  there  were,  there  would  be  a  flow 
of  the  charges.)  A  special  case  of  a  conductor  is  the  earth  ; 
and  so  there  is  no  difference  of  potential  between  any  two 
points  of  it,  if  the  charges  are  at  rest.  Owing,  however,  to 
its  immense  size  compared  with  the  conductors  which  are 
ordinarily  used,  any  variations  of  the  charges  on  the  earth 
due  to  effects  produced  at  a  given  point  on  its  surface  are 
too  minute  to  be  observed.  So  it  may  be  said  that  there 
is  never  any  change  in  the  electrostatic  condition  of 
the  earth.  Further,  since  the  remote  side  of  the  earth  (still 
at  the  same  potential  as  all  nearer  points)  is  so  far  away, 
it  may  be  said  that  there  is  no  difference  of  potential 
between  it  and  infinity,  i.  e.  a  point  at  an  infinite  distance 
from  us. 

Consequently,  it  is  convenient  to  measure  all  differences 
of  potential  from  the  earth  to  various  points,  that  is,  to  use 
the  earth  as  the  starting-point.  And  the  numerical  value 
of  the  electric  "potential  at  a  point"  is  defined  to  be  the 
amount  of  work  required  to  carry  a  unit  positive  charge 
from  the  earth  to  that  point,  or  from  infinity  to  that  point. 
This  is  equivalent  to  calling  the  electric  potential  of  the 


286  THEORY  OF  PHYSICS  [CH.  II 

earth  and  of  infinity  zero  ;  that  is,  the  electric  potential 
of  the  earth  is  taken  as  a  fixed  point.  (This  is  done  arbi- 
trarily, just  as  the  temperature  of  melting  ice  is  called  0° 
in  the  Centigrade  system.) 

226.  Potential  of  a  Conductor.     The  potential  of  all  points 
of  a  conductor,  inside  and  on   the  surface,  must  be  the 
same,  if  the  charges  are  at  rest ;  for,  if  there  was  a  differ- 
ence of  potential,  there  would  be  a  force  in  the  conductor, 
which  is  impossible,  since  the  charges  are  at  rest. 

Further,  the  potential  at  any  point  inside  a  hollow  closed 
conductor  is  the  same  as  at  a  point  of  the  conductor  itself. 
For  it  has  been  proved  (Art.  220)  that  there  are  no  charges 
inside  a  hollow  closed  conductor,  and  that  consequently 
there  are  no  lines  of  force  inside.  And,  if  there  was  a  dif- 
ference of  potential  between  a  point  in  the  hollow  space 
and  a  point  of  the  conductor,  there  would  be  a  force  be- 
tween them ;  and  so  a  line  of  force  could  be  drawn.  But 
this  has  just  been  shown  to  be  impossible.  (If  there  is  a 
second  charged  body  placed  in  the  hollow  space,  but  insu- 
lated from  the  conductor,  this  last  statement  does  not 
apply.) 

As  will  be  shown  later,  the  potentials  of  all  points  of  a 
conductor  are  not  in  all  cases  the  same,  even  if  the  charge  is 
at  rest.  If  the  temperature  of  a  charged  conductor  is  not 
the  same  at  all  points,  or  if  two  conductors  of  different 
materials  are  in  contact,  the  potential  is  different  at  differ- 
ent points  ;  but  these  variations  are  exceedingly  minute  in 
comparison  with  the  potentials  of  ordinary  charged  bodies. 

227.  Distribution    of   Charges    on    Conductors.     As    just 
stated,  when  a  charge  is  placed  at  any  point  on  a  con- 
ductor, it  distributes  itself  over  the  surface  in  such  a  way 
that  the  potential  of  the  surface  is  the  same  at  all  points, 
and  so  that  there  is  no  force  at  any  point  inside  the  outer 
surface.     The  charge  per  square-centimetre  of  the  surface, 
what  may  be  called  the  "  surf  ace-  density  "  of  electricity,  is 
by  no  means  the  same  all  over  the  surface.     It  is  much 


228]  ELECTRIC   POTENTIAL  AND  INDUCTION  287 

greater  at  sharp  points  and  edges  than  on  flat  surfaces. 
This  is  made  evident  by  the  ease  with  which  charges  pass 
from  points  off  to  the  air.  In  the  neighborhood  of  points, 
where  the  surface  density  is  great,  the  electric  force  must 
be  large ;  and,  if  it  is  sufficient  to  strain  the  air  near  the 
points  so  that  it  breaks  down,  a  minute  spark  is  caused, 
and  some  of  the  charge  passes  from  the  sharp  point  of  the 
conductor  to  the  air.  The  air  is  now  charged  with  the 
same  kind  of  electricity  as  the  conductor,  and  so  the  two 
repel  each  other.  On  a  smooth  sphere  the  surface-density 
would  be  the  same  at  all  points. 

228.  Equipotential  Surfaces.  There  must  be  in  any  elec- 
tric field  many  points  where  the  potential  is  the  same ;  in 
other  words,  such  points  that  no  work  is  required  to  carry  a 
charge  between  them.  The  locus  of  all  points  whose  poten- 
tials are  the  same  is  called  an  "  equipotential  surface." 
(Compare  the  fact  that  no  work  is  done  against  gravitation 
in  moving  any  body  along  a  horizontal  table.  The  table  is 
then  a  gravitation  equipotential  surface.)  It  is  possible  to 
construct  a  series  of  equipotential  surfaces,  corresponding 
to  different  values  of  the  potential,  each  value  being  a  con- 
stant for  one  surface.  Thus,  the  surface  of  a  charged  con- 
ductor is  an  equipotential  surface,  if  the  charge  is  at  rest. 
Again,  if  the  charge  is  on  a  spherical  conductor,  the  equi- 
potential surfaces  are  evidently,  by  symmetry,  concentric 
spherical  surfaces. 

The  direction  of  the  electric  force  at  any  point  of  an 
equipotential  surface  must  be  perpendicular  to  it.  If  it 
was  not,  there  would  be  a  component  of  the  force  in  the 
surface ;  and  in  carrying  charges  along  the  surface,  work 
would  be  required  in  order  to  overcome  this  component. 
But  this  is  impossible,  because  by  definition  there  is  no 
difference  of  potential  in  the  surface.  Consequently,  lines 
of  force  cut  the  equipotential  surfaces  at  right  angles. 
Sections  of  the  equipotential  surfaces  may  thus  be  drawn 
for  all  the  charges  represented  in  Article  222,  by  simply 


288 


THEORY  OF  PHYSICS 


[CH.  II 


drawing  closed  curves  perpendicular  to  the  lines  of  force  as 

given. 

Further,  the  direction  of   the   lines  of  force  is  always 

from  high  to  low  potential.     Let,  in  the  figure,  Fi  and  F2 

be  partial  sections  of  equi- 
potential  surfaces,  and  /  be 
the  line  of  force  perpendicu- 
lar to  them.  In  carrying  a 
positive  charge  from  F2  to 
Fi  along  the  line  of  force, 
work  must  be  done,  if  the 
direction  of  the  force  is  from 
Fi  to  F2.  Hence  Fi  -  F2  is 
positive  ;  and  so  Fi  is  greater 
than  F2. 
229.  Induction.  If  there  is  a  positive  charge  producing 

the  field,  the  potential  at  a  point  A,  which  is  near  it,  must 

be  greater  than  that  at  a  point  B,  farther  away,  because 


A  B 

FIG.  155. 

work  would  be  required  to  carry  a  positive  charge  from  B 
to  A.  If  an  uncharged  conductor  of  any  shape  is  now 
placed  connecting  A  and  B,  there  must  be  some  change  so 
as  to  make  the  potential  of  A  and  B  the  same.  The  po- 
tential at  A  must  be  lowered  and  that  at  B  raised.  This 
can  be  done  only  by  the  appearance  of  a  negative  charge  at 
A  and  a  positive  one  at  B.  As  noted  before,  any  neutral 
conductor  may  be  considered  as  having  on  it  equal  amounts 
of  positive  and  negative  electricity ;  and  so  the  amount 
near  A  is  equal  and  opposite  to  the  amount  near  B. 

Similarly,  if  the  field  is  produced  by  a  negative  charge, 
the  potential  at  a  point  A,  near  the  charge,  is  less  than  at  a 


229]  ELECTRIC  POTENTIAL  AND  INDUCTION  289 

point  J?,  farther  away,  because  work  is  required  to  carry  a 
positive  charge  from  A  to  B.  And  if  an  uncharged  con- 
ductor is  placed  so  as  to  connect  A  and  B,  the  potential  of  A 
must  be  raised  by  the  appearance  of  a  positive  charge  there, 
and  that  at  B  lowered  by  an  equal  negative  charge. 


A  B 

FIG.  156. 

These  charges  which  thus  appear  on  uncharged  conduc- 
tors when  they  are  placed  in  electric  fields  are  called  "  in- 
duced "  charges ;  and  the  entire  phenomenon  is  called 
"  electrostatic  induction." 

The  accompanying  drawing  gives  the  field  of  force  after 
an  uncharged  spherical  conductor  is  placed  in  a  uniform 
field. 


FIG.  157. 

Where  the  lines  of  force  end  on  the  conductor,  there  is  a 
negative  charge ;  and,  where  they  leave  the  surface,  there 
is  a  positive  charge.  Where  the  lines  of  force  are  most 
numerous,  the  field  is,  of  course,  most  intense. 

A  drawing  is  also  given  of  the  change  produced  in  a 
uniform  field  of  force  in  air  by  the  insertion  in  it  of  a 
sphere  of  some  dielectric  like  glass  or  paraffin,  for  which 
K  is  greater  than  for  air. 

10 


290 


THEORY  OF  PHYSICS 


[CH.  II 


FIG.  158. 

The  field  of  force  inside  the  sphere  is  also  uniform  (it 
would  be  so  also  for  any  ellipsoid  placed  with  its  axis 
parallel  to  the  original  field).  The  field  of  force  is  more 
intense  just  outside  the  sphere  than  at  a  long  distance 
away;  so  any  small  pieces  of  uncharged  conductors  or 
dielectrics  would  be  attracted  up  towards  the  sphere.  (See 
Art.  221.) 

It  requires  energy  of  course  to  produce  these  induced 
charges ;  and  the  work  is  done  by  the  external  force  which 
brings  the  uncharged  conductor  into  the  field  of  force,  or 
which  brings  the  field  of  force  near  the  uncharged  body. 

230.   Illustrations   of  Induction.     The  fact  that  when   a 
charged  body   is   brought   near   a   gold-leaf   electroscope, 
,        changes  in  the  position  of  the 
leaves  are  produced,  is  at  once 
explained  as  due  to  the  induced 
charges   caused   by   the   charge 
which  is  carried  near. 

Again,  the  reason  may  be  given 
why  a  conductor  which  has  on 
its  surface  sharp  points  will  dis- 
charge any  charged  body  near  the 
points,  and  will  itself  become 
similarly  charged.  The  charged 
body  induces  charges  in  the  con- 
ductor, —  an  opposite  charge  on 
the  pointed  side  nearest  it  and  a  similar  charge  on  the 
further  side.  As  previously  explained,  the  charges  on  the 


FIG.  159. 


232]  ELECTRIC  POTENTIAL  AND  INDUCTION  291 

points  may  easily  pass  off  to  the  air ;  and  then  these 
charged  particles  in  the  air  will  be  attracted  by  the  origi- 
nally charged  body,  since  the  charges  are  opposite.  So  the 
latter  body  becomes  discharged,  while  a  charge  has  re- 
mained on  the  pointed  conductor. 

231.  "  Charging  by  Induction."     If  a  conductor  is  placed 
in  a  field  of  force,  its  potential  will  be  that  of  the  field  at 
the  point  where  it  is  placed ;  and  this  will  be,  in  general, 
either  positive  or  negative,  that  is,  greater  or  less  than  that 
of  the  earth.     So,  if  the  conductor  is  now  joined  to  the 
earth  by  some  conductor,  e.  g.  a  wire  or  the  human  body, 
its  potential  must  immediately  become  0.     If  its  potential 
was  +  before,  then  a  negative  charge  must  appear  on  it  so 
as  to  lower  the  potential  (the  equal  negative  charge  is  on 
the  earth).     If  its  potential  was  —  before,  a  positive  charge 
will  appear  on  it  so  as  to  raise  its  potential  to  0.     (This 
case  is  just  like  the  one  treated  in  the  previous  section, 
Article  229,  if  the  point  B  is  taken  on  the  earth,  and  the 
conductor  is  supposed  to  consist  of  the  original  uncharged 
conductor,  the  wire  and  the  earth.)     If  now  the  connec- 
tion with  the  earth  is  broken,  the  conductor  which  was 
originally  uncharged  is  charged  either  positively  or  nega- 
tively, depending  upon  its  original  potential.      This  pro- 
cess of    producing  charges  on  bodies  is  called  "  charging 
by  induction." 

232.  Faraday's  "  Ice-Pail  Experiment."     A  famous  experi- 
ment in  induction,  arid  one  of  great  theoretical  importance, 
was  performed  by  Faraday  by  means  of  an  ice-pail  and  a 
small  charged  conductor.     It  may  be  repeated,  using  in- 
stead of  the  ice-pail  a  hollow  conductor,  which  is  more 
nearly   closed.      The   nearly   closed   hollow   conductor  is 
insulated  from  the  earth,  and  is  joined  by  a  wire  to  an 
electroscope,  such  as  a  gold-leaf  one.     The  conductor  is 
uncharged,  and  so  there  is  no  diverging  of  the  gold-leaves. 
If  a  small  charged  conductor  is  now  carefully  lowered  by 
means    of  a  silk    thread  into  the  interior  of   the  hollow 


292 


THEOBY  OF  PHYSICS 


[CH.  II 


conductor,  so  as  not  to  touch  it  at  any  point,  charges  are, 
of  course,  induced  on  the  inside  and  outside  of  the  hollow 
conductor.  Consequently,  the  gold-leaves  of  the  electro- 
scope will  diverge.  Two  interesting  facts  may  be  observed : 
(1)  if  the  small  charged  conductor  is  moved  at  random 
inside  the  hollow  conductor,  but  not  touching  it,  there  is 


FIG.  160. 

no  change  in  the  divergence  of  the  gold-leaves ;  (2)  if  the 
small  charged  conductor  is  allowed  to  strike  the  inside  of 
the  hollow  conductor,  there  is  still  no  change  in  the  leaves. 
This  proves  that  the  small  charged  body  induces  on  the 
inside  and  outside  of  the  practically  closed  conductor  around 
it  charges  equal  to  itself ,  —  one  opposite,  the  other  similar. 
The  two  induced  charges  must,  of  course,  be  equal  to  each 
other  but  of  opposite  kinds.  The  charge  on  the  inside 
must  be  of  an  opposite  kind  to  that  on  the  small  body 
which  is  inserted ;  and  these  two  must  be  equal ;  because, 


233]  ELECTRIC  POTENTIAL  AND   INDUCTION  293 

when  the  two  conductors  touch,  there  is  no  change  on  the 
outside  (as  proved  by  the  absence  of  change  of  the  gold- 
leaves),  and  so  the  charge  on  the  small  conductor  must 
exactly  neutralize  that  on  the  inside  of  the  hollow 
conductor. 

233.  "  Faraday  Tubes."  Faraday  proposed  a  description 
of  these  phenomena  of  induction,  which  has  the  great  ad- 
vantage of  presenting  a  clear  picture  of  them  to  the  mind. 
He  thought  of  there  being 
tubes  constructed  through- 
out an  electric  field,  these 
tubes  being  built  up  of 
lines  of  force ;  that  is,  a 
closed  curve  is  taken  in 

the  field  of  force,  and  lines  of  force  are  drawn  through 
each  point  of  the  curved  line,  thus  making  a  hollow  tube. 
Each  tube  is  made  of  such  dimensions  that,  when  it 
reaches  a  charged  body,  its  end  shall  exactly  include  a 
unit  charge.  Of  course,  different  tubes  have  different  sizes  ; 
and  the  cross  section  of  any  one  tube  varies,  except  in  a 
uniform  field,  where  the  tubes  are  of  the  same  cross-section 
throughout. 

Thus,  a  body  charged  with  6  units  has  six  tubes  leaving 
it,  if  it  is  + ;  or  ending  on  it,  if  it  is  negative.  And  the 
quantity  of  the  charge  on  any  body  may  be  known  if  the 
number  of  tubes  ending  on  it  or  starting  from  it  is  known. 
In  particular,  when  an  uncharged  conductor  entirely  en- 
closes a  body  which  is  charged  with  a  quantity  +  e,  e  tubes 
leave  the  inner  body  and  end  on  the  inner  side  of  the 
enclosing  conductor.  Hence  there  is  a  charge  —  e  on  the . 
inside  of  this  conductor.  But  since  this  conductor  was 
originally  uncharged,  as  many  tubes  must  leave  it  as  end 
on  it ;  and  so  +  e  tubes  must  leave  the  outer  surface,  show- 
ing that  there  is  a  charge  +  e  there. 

Again,  where  the  surface  density  of  a  charge  on  a  con- 
ductor is  greatest,  there  must  be  the  greatest  number  of 
Faraday  tubes. 


294  THEORY  OF  PHYSICS  [CH.  II 

234.  Shielding  by  Closed  Conductors.     As  stated  above,  in 
Article  226,  the  potential  of  a  closed  conductor  is  the  same 
at  all  points  inside,  so  far  as  any  charges  on  the  outside  of 
the  conductor  are  concerned.     So  there  is  no  electric  force 
in  the  interior ;  for,  if  there  was,  there  would  be  a  differ- 
ence of  potential.     Consequently,  no  matter  what  electric 
changes  go  on  outside  a  closed  conductor,  there  is  no  corre- 
sponding change  inside ;  so  that  the  interior  is  entirely 
shielded  from  external  effects.     (This  is  not  true  of  the 
magnetic  effect  produced  by  electric  currents.) 

Again,  if  there  are  any  charged  bodies  in  the  interior  of 
'a  closed  conductor,  there  will  be  charges  induced  on  both 
the  inner  and  outer  surface  of  the  conductor ;  and  so  there 
will  be  a  field  of  force  outside,  which  will  change  if  the 
charges  inside  are  changed.  But,  if  the  conductor  is  joined 
to  the  earth  by  a  wire  or  other  conductor,  the  field  of 
force  outside  vanishes ;  because,  now  both  the  conductor 
and  the  ear.th  are  at  the  same  potential,  and  so  there  are 
no  lines  of  force  in  the  field.  (A  line  of  force  must  pass 
from  high  to  low  potentials.)  Consequently  a  hollow  closed 
conductor  joined  to  the  earth  completely  shields  the  ex- 
terior space  from  any  effects  due  to  charges  inside. 

Induction-Machines.  Various  devices  have  been  made  by 
means  of  which  to  utilize  the  method  of  charging  by  in- 
duction so  as  to  secure  unlimited  amounts  of  electricity. 
These  are  called  "induction-machines."  The  oldest  one 
is  known  as  the  "  electrophorus,"  and  was  invented  by 
Volta.  Later  forms  are  the  Voss  machine  and  the  Wims- 
hurst  machine.  Full  descriptions  of  these  are  given  in 
larger  treatises  on  Experimental  Physics.  The  principle 
made  use  of  in  them  all  is  to  turn  mechanical  energy  into 
electrical  by  using  mechanical  power  to  produce  induced 
charges. 

235.  Condensers.     If  a  conductor  is  placed  in  air  at  some 
distance  away  from  other  conductors,  and  if  it  is  given  a 
succession  of  positive  charges,  its  potential  wil^  rise,  be- 


235]  ELECTKIC   POTENTIAL  AND   INDUCTION  295 

cause  more  and  more  work  will  be  required  to  carry  a 
positive  charge  from  the  earth  to  it.  (Similarly,  if  it  was 
given  a  constantly  increasing  negative  charge,  its  potential 
would  fall.)  A  limit  will  finally  be  reached,  though,  when 
for  a  certain  charge  the  difference  of  potential  between  the 
conductor  and  the  earth  (or  some  other  conductor)  is  so 
great  that  a  spark  will  pass ;  that  is,  the  surrounding  die- 
lectric is  so  strained  that  it  yields  to  the  stress  and 
"  breaks  down."  If,  however,  the  potential  of  the  charged 
conductor  could  be  made  nearer  that  of  the  earth  in  any 
way,  the  strain  would  be  lessened ;  and  so  the  conductor 
could  contain  a  greater  charge  before  the  medium  would 
break. 

This  can  actually  be  done.     Let  A  be  a  metal  plate 
charged  positively,  and  let  its  potential  be  V\  the  problem 
is  to  see  how  this  potential  can  be  diminished       / 
without  decreasing  the  charge  on  the  plate. 
Bring  near  A    an  insulated   uncharged   con- 
ductor B,  such  as  another  metal  plate ;  charges 
are  induced  on  B,  negative  on  the  side  towards 
A,  positive   on  the  further  side.     If  a  unit 
positive  charge  is  now  brought  up  to  A  from 
the  earth,  less  work  is  required  than  was  be-    +   -f-     ! 
fore  B  was  in  place,  because  now  most  of  the 
strain  in  the  medium  is  between  A  and  B  (or,      FlG>  162. 
in  terms  of  another  description,  because  the 
negative  charge  on  B,  being  nearer  A  than  the  positive 
charge  is,  helps  in  doing  the  work  by  its  attraction  on 
the  positive  charge  which  is  being  brought  up  from  the 
earth).     Consequently  the  potential  of  A  has  been  low- 
ered.    It  may  be  lowered  still  further  if  the  conductor 
B  is  joined  to  the  earth  by  a  wire  or  other  conductor, 
because  now  still  more  of  the  strain  is  drawn  into  the 
space  between  A  and  B  (or  because  the  positive  charge 
on  B  now  goes  to  the  earth);  and  so  less  work  would 
be  required  to  bring  up  a  unit  positive  charge  from  the 


296 


THEORY  OF  PHYSICS 


[CH.  II 


earth.  The  potential  of  the  conductor  B  is  now  0,  the 
same  as  that  of  the  earth  ;  and  that  of  the  conductor 
A  equals  the  amount  of  work  required  to  carry  a  unit 
positive  charge  from  B  to  A.  If  a  plate  of  any  dielectric 
for  which  K  is  greater  than  air  is  now  inserted  between  A 
and  B  (e.  g.  a  plate  of  glass  or  of  paraffin),  replacing  the 
air,  less  work  would  be  required  to  carry  a  unit  positive 
charge  from  B  to  A,  because  in  glass  or  paraffin  electric 
forces  are  less  than  in  air,  there  is  less  strain  than  in  air. 
So  the  potential  of  A  is  again  lowered.  To  sum  up,  then, 
the  potential  of  A  may  be  lowered  by  bringing  near  it 
another  conductor  connected  to  the  earth  and  separated 
from  A  by  a  dielectric  for  which  K  is  as  great  as  conven- 
ient. So  that  now  the  conductor  A  can  receive  a  much 
greater  positive  charge  than  it  could  when  by  itself,  before 
its  potential  is  raised  to  any  definite  value. 

In  an  exactly  similar  manner  it  may  be  proved  that,  if  A 
is  charged  negatively  and  so  has  a  negative  potential,  its 
potential  may  be  raised,  i.  e.  brought  nearer  that  of  the 
earth,  by  bringing  near  it  a  conductor  B  joined  to  the  earth 

and  separated  from  A  by  a  die- 
lectric, for  which  K  is  very  large. 
So  the  conductor  A  must  now 
receive  a  much  greater  negative 
charge  than  it  had  before,  if  its 
potential  is  to  be  as  low. 

Such  an  arrangement,  two  con- 
ductors separated  by  a  dielectric, 
is  called  a  "  condenser."  The 
commonest  type  of  condenser  is 
one  where  the  conductors  are 
parallel  plates  ;  these  may  be  flat 
pieces  of  tin-foil  separated  by  flat 
pieces  of  glass  or  paraffined  paper ;  or  they  may  be  curved, 
as  in  the  Leyden  Jar,  which  is  a  glass  bottle,  coated  inside 
and  out  with  tin-foil,  as  shown.  The  tin-foil  does  not 


FIG.  163. 


236]  ELECTRIC  POTENTIAL  AND  INDUCTION  297 

extend  up  to  the  edge  of  the  bottle,  and  the  interior  foil 
is  connected  to  a  metal  ball  outside  by  means  of  a  metal 
chain. 

If  one  of  the  conductors,  B,  is  joined  to  the  earth,  its  charge 
will  be  opposite  to  that  on  A ;  and,  if  the  two  conductors 
are  near  each  other,  like  two  parallel  plates  close  together, 
or  if  one  conductor  encloses  the  other  like  two  concentric 
spheres  or  two  coaxal  cylinders,  the  two  charges  are  almost 
exactly  equal.  (In  the  case  of  two  concentric  spheres, 
they  are  equal.)  For,  practically,  all  the  Faraday  tubes 
leaving  one  conductor  end  on  the  other.  If  the  connection 
with  the  earth  is  now  broken,  and  the  entire  apparatus 
moved  elsewhere,  the  potentials  of  the  two  conductors 
may  change,  but  their  difference  will  remain"  the  same, 
because  the  same  amount  of  work  is  required  after  the 
change  in  position  as  before,  to  carry  a  unit  positive 
charge  from  one  conductor  to  the  other.  Consequently 
a  charged  condenser  may  be  considered  as  made  up  of 
two  conductors  at  different  potentials,  V\  and  F2,  carry- 
ing charges  +  e  and  e,  and  separated  by  a  dielectric 
whose  constant  is  K. 

236.  Discharge  of  Condensers.  The  medium  between  the 
two  conductors  of  a  charged  condenser  is  of  course  strained, 
and  the  discharge  of  the  condenser  consists  in  the  release  of 
this  strain.  The  simplest  method  is  to  join  the  two  con- 
ductors by  a  wire ;  for  then  their  potentials  become  the 
same,  the  two  equal  and  opposite  charges  neutralize  each 
other,  and  the  strain  disappears.  Analogy  from  mechanics 
would  lead  us  to  expect  two  kinds  of  discharge,  a  steady 
one  and  an  oscillatory  one.  If  a  wire  strained  by  twisting 
is  allowed  to  untwist  and  so  lose  its  strain,  it  can  do  so  in 
two  ways,  depending  upon  external  conditions;  if  the  wire 
is  held  rather  firmly  by  the  fingers,  but  is  allowed  to  slip 
through  them,  the  twist  disappears  slowly  and  continu- 
ously ;  if  the  wire,  though,  has  considerable  inertia  and  is 

free  to  twist,  it  will  untwist,  twist  in  the  opposite  direction, 

10* 


298  THEORY  OF  PHYSICS  [CH.  II 

untwist,  etc.,  making  a  series  of  oscillations  or  reversals, 
but  finally  coming  to  rest  as  they  die  out.  That  is,  if  there 
is  great  frictional  resistance,  the  discharge  of  the  strain  is 
continuous ;  if  there  is  great  inertia  and  no  friction,  the 
discharge  is  oscillatory. 

It  is  exactly  the  same  in  the  discharge  of  a  condenser ; 
the  medium  is  strained  and  held  so  by  definite  forces ; 
when  these  forces  are  removed,  the  discharge  takes  place. 
If  there  is  a  great  frictional  resistance  to  the  dying  out  of 
the  strain,  it  will  discharge  continuously,  the  positive 
charge  will  grow  less,  and  so  will  the  negative ;  if  there  is 
a  small  frictional  resistance,  the  strain  will  discharge  itself, 
then  become  reversed,  discharge  itself,  return  to  its  pre- 
vious type,  etc.  When  the  strain  is  reversed,  the  conduc- 
tor which  was  positively  charged  becomes  negative,  and 
the  one  which  was  negative  becomes  positive ;  but,  as  the 
strain  is  repeatedly  reversed,  the  corresponding  charges  be- 
come less  and  less,  and  so  in  the  end  the  strain  is  dis- 
charged. The  first  of  these  two  kinds  of  discharge  may  be 
obtained  by  joining  the  two  conductors  of  the  charged 
condenser  by  a  bad  conductor;  the  second,  by  joining  them 
by  a  good  conductor.  The  discharge  is  in  any  case  ex- 
tremely rapid,  and  is  studied  with  difficulty. 

In  some  condensers  it  is  observed  that,  after  being  ap- 
parently discharged  by  having  their  two  conductors  joined, 
they  become  charged  again  on  standing  some  time ;  and,  if 
this  secondary  charge  is  discharged,  there  may  be  another 
charge,  etc. ;  but  each  succeeding  discharge  is  much  less 
than  the  preceding  one.  It  has  been  proved  that  these 
secondary  discharges  occur  only  in  such  condensers  as 
have  for  their  dielectric  some  heterogeneous  material  such 
as  glass.  A  condenser  which  has  a  gas,  a  liquid,  or  a 
homogeneous  solid  like  a  pure  crystal  for  its  dielectric  has 
no  secondary  discharges.  When  a  condenser  with  a  non- 
homogeneous  dielectric  is  discharged,  the  strain  does  not 
vanish,  but  some  parts  are  strained  in  opposite  directions 


237]  ELECTRIC  POTENTIAL  AND   INDUCTION  299 

to  other  parts,  thus  producing  apparent  absence  of  strain. 
But,  if  now  the  medium  yields  to  the  strain  in  some  part, 
the  opposing  strains  are  no  longer  balanced,  and  so  there 
is  a  secondary  charge. 

237.  Capacity.  If  a  given  conductor  is  placed  by  itself 
in  some  dielectric,  at  a  great  distance  from  other  conduc- 
tors, and  is  given  a  charge  of  any  kind,  it  will  have  a 
definite  potential.  Let  the  charge  be  e,  and  the  potential  be 
V.  It  is  found  by  experiment  that  one  is  proportional  to 
the  other ;  that  is,  that,  if  there  is  twice  the  charge,  the 
potential  is  twice  as  great,  etc.  This  may  be  expressed 
mathematically  by  writing 

e=CV .     (I) 

C  is  a  constant  for  the  particular  conductor,  and  the  partic- 
ular surrounding  dielectric.  C  does  not  depend  upon  the 
material  of  the  conductor,  but  only  on  its  shape  and  size. 
It  is  called  the  "  capacity,"  and  is  evidently  the  numerical 
value  of  that  charge  which  is  required  to  produce  a  poten- 
tial 1  on  the  particular  conductor  in  the  particular  dielec- 
tric. -  It  may  be  proved  that  the  capacity  of  a  spherical 
conductor  of  radius  R  in  a  dielectric  whose  constant  is  K 
is  C=K  R. 

Similarly,  if  the  equal  and  opposite  charges  on  a  conden- 
ser are  varied,  the  corresponding  differences  of  potential 
vary  in  such  a  way  that,  if  e  is  the  charge  on  one  conductor 
(—  e  being  on  the  other)  and  Fi  —  F"2  the  corresponding 
difference  of  potential, 

e=c(r1-rt-), (2). 

where  C  is  a  constant  for  the  particular  condenser,  and  is 
called  its  capacity.  C  depends  obviously  on  the  shape  and 
size  of  the  two  conductors  and  upon  the  dielectric  between 
them. 

For  two  concentric  spheres  of  radii  RI  and  EZ)  separated 

Jf  7?    7? 

by  a  medium  whose  constant  is  K,  C  =  — — — —  . 


300  THEORY  OF  PHYSICS  [CH.  II 

For  two  parallel  plates  of  area  A,  at  a  distance  d  apart, 

K A 
and  separated  by  a  medium  whose  constant  is  K,  C  —  -r — - 

~r  7T  Ct 

(TT  =  3.1416,  the  ratio  of  the  circumference  of  a  circle  to  its 
diameter). 

If  any  quantity  of  electricity  is  given  to  two  condensers 
of  identically  the  same  dimensions,  but  having  different 


EARTH 

FIG.  164. 

dielectrics,  the  charges  will  be  divided  in  the  ratio  of  the 
values  of  K.  For  let  two  such  parallel-plate  condensers  be 
joined,  so  that  one  plate  of  each  is  connected  by  a  wire  to 
one  of  the  other  ;  and  join  one  of  these  pairs  by  a  wire  to 
the  earth.  Give  to  the  other  pair  of  plates  any  quantity,  e. 
This  will  distribute  itself  so  that  one  condenser  has  e\  ;  the 
other,  e%  where  e  =  e\ 


But  ei^iCFx 

fcrrr&CFi-   F2), 

since  the  differences  in  potential  are  the  same.  Hence  el  /  e2 
=  Oi  I  Cz  =  KI  I  Kz,  since  the  two  condensers  are  identical, 
with  the  exception  of  the  dielectrics.  Thus,  by  compar- 
ing the  values  e\  and  ez,  the  ratio  of  £\  to  K2  may  be  deter- 
mined. Some  values  of  K  are  given  in  the  following 
table  :  — 


238]  ELECTRIC  POTENTIAL  AND  INDUCTION 

TABLE  XII 

DIELECTRIC  CONSTANTS  (ELECTROSTATIC  SYSTEM). 


301 


Glass  (about)     . 
Mica    .... 

6 

8 

Turpentine    .     . 
Petroleum     .     . 

2.4 
2.1 

Paraffin     .     .     . 
Kubber     .     .     . 
Water.     .     .     . 

2 
2.5 
76 

Hydrogen      .     . 
Illuminating  Gas 
Carbon  Dioxide 

0.9998 
1.0004 
1.0008 

Alcohol     .     .     . 

26 

(Vacuum)      .     . 

0.9995 

238.  Energy.  The  energy  which  is  associated  with  elec- 
tric charges  is,  as  has  been  said  many  times,  in  the  dielec- 
tric forming  the  electric  field;  and  it  is  not  difficult  to 
calculate  the  amount  of  the  energy  per  cubic  centimetre 
at  any  point  of  the  field,  expressed  in  terms  of  the  proper- 
ties of  the  field  at  that  point.  It  is  much  easier,  however, 
to  calculate  the  entire  energy  of  the  field  in  terms  of  the 
charges  and  their  potentials. 

If  there  is  a  conductor  whose  charge  is  e  and  whose 
potential  is  V,  the  work  required  to  produce  this  charge  is 
\e  V.  The  charging  may  be  considered  as  having  been 
done  gradually,  so  that  the  potential  has  slowly  risen  from 
0  to  F.  If  the  potential  at  any  instant  is  V,  the  work 
required  to  bring  up  from  the  earth  a  unit  positive  charge 
is  F',  if  the  potential  does  not  change  ;  and  so  the  work 
necessary  to  bring  up  a  charge  +  e>  would  be  e  F',  if  the 
potential  did  not  change.  But  while  e  is  being  brought  up 
in  small  equal  amounts,  the  potential  rises  regularly  from 
0  to  F,  and  so  the  average  potential  is  J  F.  The  entire 
work  done,  then,  in  bringing  up  the  charge  +  e  to  the  aver- 
age potential  J  F  is  the  product  of  the  two,  J  e  F.  Since 
e=CV}  this  may  be  expressed 

energy  =  }«r=J0F»  =  j£     .     .     .     (3) 

(The  negative  charge  —  e  is  on  the  earth  at  the  potential  0 ; 
so  this  may  be  said  to  be  the  work  required  to  separate  the 


302  THEORY  OF  PHYSICS  [CH.  II 

charges  +  «  and  —  e  until  their  difference  of  potential  is 

v.) 

The  charges  on  a  condenser  are  +  e,  at  a  potential  V\  and 
—  e  at  a  potential  F"2.  Hence  the  work  required  to  produce 
these  charges,  that  is,  the  entire  energy  of  the  field,  is 


. 
this  may  be  expressed, 

energy  =  £e  (FI  -  F2)  =  |-  C  (  F,  -  F2)2  =  j£.    .     (4) 

U 

239.  Electrostatic  Measurements.  There  are  three  quan- 
tities which  must  be  measured  in  order  to  express  the 
electric  properties  of  a  charged  conductor,  —  quantity  of 
charge,  potential,  capacity. 

Capacity  depends  upon  the  size  and  shape  of  conductor 
and  the  constant  of  the  dielectric  ;  and  in  certain  simple 
cases  its  value  may  be  calculated,  as  indicated  in  Article 
237. 

Potential,  or  difference  of  potential,  may  be  measured 
by  certain  instruments  known  as  "  electrometers."  One 
form  of  instrument  depends  upon  the  fact  that  the  two 
plates  of  a  parallel-plate  condenser  tend  to  approach  each 
other  because  one  is  charged  positively  and  the  other  nega- 
tively. It  may  be  proved  that,  if 

Vi  —  F"2  =  difference  of  potential  of  the  two  plates, 
d  =  the  distance  apart  of  the  two  plates, 
A  =  area  of  a  movable  portion  of  one  plate  near 
its  centre, 

the  mechanical  force  which  must  be  applied  to  this  mov- 
able disc  to  keep  it  from  moving  towards  the  other  plate  is 


Hence  Fi  -  V,  =  d  V  ^j (5) 


239]  ELECTRIC  POTENTIAL  AND  INDUCTION  303 

If  air  is  the  dielectric,  K  =  1  on  the  electrostatic  system ; 
and  it  is  possible  to  measure  F,  d  and  A  ;  so  V\  —  F2  may 
be  calculated. 

A  disc  is  taken  near  the  centre  of  one  plate,  so  as  to 
have  a  region  where  the  field  is  uniform.  If  the  force  on 
the  entire  plate  was  measured,  it  would  be  necessary  to 
make  a  correction  for  the  lack  of  uniformity  at  the  edges. 

Knowing  capacity  and  difference  of  potential,  quantity 
may  be  calculated.  There  is  a  method  which  will  be  de- 
scribed later  on  by  means  of  which  quantity  can  be  meas- 
ured, but  not  in  terms  of  the  electrostatic  system. 


CHAPTER   III 

ELECTRIC    CONDUCTION 

IN  the  previous  chapters,  the  general  properties  and 
laws  of  electric  charges  at  rest  have  been  considered.  But 
it  is  possible  for  the  charges  to  move,  i.  e.  for  the  electric 
strain  to  change ;  and  in  this  and  the  following  chapters 
will  be  studied  the  different  methods  of  producing  motion 
of  the  charges,  and  the  laws  of  the  various  phenomena 
associated  with  this  motion. 

240.  Electric  Current.  If  a  series  of  small  charged  bodies 
move  in  a  given  line,  or  if  a  charge  passes  along  a  conduc- 
tor, this  phenomenon  is  called  an  electric  "  current."  If  a 
charge  +  e  goes  by  a  fixed  point  in  one  direction  in  one  sec- 
ond, and  if  in  the  same  time  a  charge  —  e'  goes  by  in  the 
opposite  direction,  the  sum,  e  +  ef,  is  called  the  "  intensity  " 
of  the  current,  and  its  direction  is  said  to  be  that  in  which 
the  positive  charge  moves.  If  the  current  is  steady,  the 
quantity  of  electricity  carried  by  in  t  seconds  is  the  pro- 
duct of  t  and  the  intensity  of  the  current. 

The  simplest  type  of  a  current  in  a  conductor  is  that 
afforded  when  two  plates  of  a  charged  condenser  are  joined 
by  a  conductor,  e.  g.  a  fine  wire,  as  shown.  The  charges  on 
the  two  plates  become  less  and  less,  as  is  manifest  by  the 
vanishing  of  the  strain  in  the  medium  between  the  two 
plates.  While  this  process  is  going  on,  the  wire  joining 
the  two  plates  grows  warm,  showing  that  its  molecules  are 
receiving  energy;  other  effects  are  also  produced,  which 
will  be  discussed  later ;  and  the  existence  of  a  current 
can  always  be  proved  by  any  one  of  these  effects.  The 


240] 


ELECTRIC   CONDUCTION 


305 


energy  of  the  dielectric  has  thus  passed  into  the  wire ;  and 
there  is  evidence  for  believing  that  it  passes  into  the  wire 
perpendicularly,  as  shown  by  the  dotted  lines. 


FIG.  165. 

There  is  said  to  be  an  electric  current  in  the  wire  join- 
ing the  plates,  as  long  as  the  energy  continues  to  come 
into  the  wire  from  the  dielectric.  Unless  the  charges  on 
the  plates  are  renewed,  this  process  will  soon  stop.  But 
if,  as  fast  as  the  strain  dies  down,  it  is  restored,  the  pro- 
cess will  continue  unchanged ;  and  there  is  said  to  be  a 
"steady"  current  in  the  wire.  This  may  be  expressed 
mathematically  by  saying  that,  if  the  difference  of  poten- 
tial, V\  —  Vz,  can  be  maintained  constant,  there  will  be  a 
steady  current,  and  its  direction  in  the  connecting  wire 
will  be  from  the  point  of  high  potential  to  that  of  low, 
i.  e.  from  V\  to  V2.  (Notice  the  analogy  of  the  flow  of 
heat  energy  from  high  temperature  to  low,  and  of  the  flow 
of  volume  energy  from  high  pressure  to  low.  In  this  case 
electric  energy  passes  from  high  to  low  potential.)  Wher- 
ever there  is  a  difference  of  potential,  there  is  an  electric 
force  from  high  to  low ;  that  is,  a  positive  charge,  if  free 
to  move,  would  pass  from  high  to  low  potential,  while  a 
negative  charge  would  pass  in  the^opposite  direction.  So 


306  THEORY  OF  PHYSICS  [CH.  Ill 

in  this  case  of  the  condenser,  since  a  conductor  joins  the 
two  plates,  and  since  they  may  be  considered  as  having 
equal  amounts  of  positive  and  negative  charges,  the  former 
will  move  in  one  direction  in  the  wire,  the  latter  in  the 
opposite.  While  the  charges  are  moving  there  is  no  reason 
for  believing  that  they  are  on  the  surface  of  the  conductor ; 
and,  in  fact,  it  is  known  that,  except  in  the  case  of  rapidly 
alternating  currents,  the  current  is  inside  the  conductor  as 
well  as  on  the  surface. 

The  one  necessary  condition,  therefore,  for  .the  production 
of  a  current  in  a  conductor  is  some  means  of  keeping  its 
two  ends  at  a  difference  of  potential :  if  this  difference  is 
constant,  the  current  is  steady ;  otherwise,  not.  A  name 
has  been  given  to  the  difference  of  potential  between  any 
two  points  of  a  conductor  through  which  a  current  is  flow- 
ing, viz.,  the  "  electro-motive  force  "  (E.  M.  F.)  between  those 
points. 

There  are  many  ways  by  which  a  continuous  E.  M.  F.  may 
be  produced  and  maintained.  The  simplest  method  is  to 
use  an  electric  machine,  either  a  friction  or  an  induction 
one,  and  to  join  its  two  "  poles  "  to  the  conductor  in  which 
a  current  is  desired. 

Another  method  is  to  make  a  circuit,  part  of  one  con- 
ductor, part  of  another ;  and  to  keep  the  two  junctions  at 
different  temperatures.  It  is  observed  that  under  these 
conditions  there  is,  except  in  special  cases,  a  current,  which 
is  called  a  "  thermo-electric  current." 

Again,  it  is  observed  that,  if  two  different  solid  conduc- 
tors dip  into  a  liquid  conductor,  there  is  an  E.  M.  F.  between 
the  ends  of  the  solids,  which  rise  above  the  liquid.  So,  if 
they  are  joined  by  a  conductor,  e.  g.  a  wire,  there  will  be  a 
current  in  it. 

Still  another  method  depends  upon  certain  magnetic 
properties,  and  will  be  discussed  later. 

In  all  these  cases  energy  passes  from  the  dielectric  into 
the  conductor,  in  which  the  current  is  "  flowing ; "  and  this 


241] 


ELECTRIC  CONDUCTION 


307 


energy  must  be  maintained  by  a  supply  of  energy  from 
some  source.  In  the  case  of  both  the  electric  and  mag- 
netic machines,  the  work  is  done  by  some  external  body 
turning  the  machine.  In  thermo-electric  currents  the 
energy  comes  from  the  bodies  which  maintain  the  differ- 
ences in  temperature  at  the  junctions.  Where  the  two 
solid  conductors  dip  in  a  liquid  one,  the  energy  comes  from 
certain  chemical  reactions. 

241.  Thermo-Electric  Currents.  The  conditions  for  a 
thermo-electric  current  are :  a  circuit  made  of  two  differ- 
ent conductors  1  and  2,  having  junctions  at  A  and  B ;  and 
the  maintenance  of  different  temperatures  at  these  junc- 
tions. The  explanation  of  the  cause  of  the  current  is  not 
difficult. 

B 


FIG.  166. 

If  any  two  conductors,  1  and  2,  are  placed  in  contact  at 
A,  they  do  not  come  to  the  same  potential  exactly  (see 
Art.  226).  Call  the  potential  of  conductor  1,  Fi ;  and  that 
of  2,  F2 ;  and  let  FI  >  F2.  The  natural  tendency  of  all 
conductors  in  contact  is  to  come  to  the  same  potential ;  but, 
since  V\  is  higher  than  F2,  there  must  be  some  electric 
force  acting  from  2  towards  1,  across  A,  so  as  to  oppose  the 
tendency  of  Fi  to  become  as  small  as  F2.  A  mechanical 
analogy  may  make  the  matter  clearer.  If  at  the  bottom  of 
a  U-tube  half  filled  with  water  there  is  a  paddle-wheel,  so 


308 


THEORY   OF  PHYSICS 


[CH.  Ill 


1 

" 


FIG.  167. 


arranged  as  to  drive  the  water  in  one  direction  or  the  other ; 
and,  if  around  the  axle  of  this  wheel  a  rope  is  coiled,  at 
whose  free  end  hangs  a  heavy  weight, 
—  the  wheel  will  turn,  thus  forcing  the 
water  up  in  one  arrn  of  the  tube  to  a 
greater  height  than  it  is  in  the  other. 
The  wheel  will  come  to  rest  when  the 
back-pressure  due  to  the  difference  in 
pressure  on  the  two  sides  of  the  wheel 
exactly  equals  the  pressure  which  the 
wheel  exerts  owing  to  the  heavy  weight. 
The  pressure  at  the  side  1,  pi,  is  greater 
than  that  at  the  side  2,  p2,  i.  e.  pi  >  pz. 
These  two  pressures  tend  to  become 
equal ;  but  are  kept  from  so  doing  by  a 
pressure  at  A,  produced  by  the  falling 
weight,  which  acts  from  2  towards  1 
through  A. 

In  the  case  of  the  two  conductors  1  and  2,  this  force 
acting  from  2  towards  1,  i.  e.  tending  to  move  a  positive 
charge  in  that  direction  and  a  negative  one  in  the  other, 
must  be  produced  by  the  differences  in  the  "  specific 
attractions "  of  the  two  kinds  of  matter  for  electricity. 
(See  Art.  217.)  Its  existence  may  be  easily  proved.  If 
a  current  is  passed  through  the  junction  A  from  1  to  2 
by  means  of  some  electric  machine  (say),  i.  e.  if  a  positive 
charge  is  forced  from  1  to  2,  and  a  negative  one  from  2  to  1, 
resistance  will  be  experienced  at  the  junction  A  if  there  is 
a  force  there  acting  from  2  to  1 ;  work  will  have  to  be  done 
there  against  "  molecular  forces ; "  energy  will  be  given  the 
molecules  there ;  and  so  a  heat  effect  which  corresponds  to 
addition  of  heat  energy  should  be  observed  at  the  junction. 
The  difference  of  potential  is  Vl  —  F2,  or  E.  So  this  is  the 
work  which  would  be  done  if  a  unit  positive  charge  was 
carried  across  from  1  to  2,  or  if  a  unit  negative  charge  was 
carried  from  2  to  1.  If,  then,  a  current  whose  intensity  is 


241]  ELECTRIC  CONDUCTION  309 

i,  is  carried  across  from  1  to  2  for  t  seconds,  i.  e.  if  a  quan- 
tity i  t  passes,  the  work  done  at  the  junction  is  E  it ;  and 
this  work  is  spent  in  adding  heat  energy  to  the  junction, 
since  V\  >  F"2.  Consequently,  its  temperature  will  rise, 
and  this  fact  can  be  observed. 

If,  on  the  other  hand,  a  current  is  forced  through  A 
from  2  towards  1,  the  electric  force  at  the  junction  helps  the 
current  on,  it  itself  does  work  ;  and  so  the  molecules  there 
lose  energy,  and  the  temperature  of  the  junction  falls.  The 
loss  in  energy,  if  a  current  whose  intensity  is  i  passes  for  t 
seconds,  is  E  it ;  the  same  as  the  energy  gained  when  the 
current  is  in  the  opposite  direction. 

Consequently,  if  on  passing  a  current  through  any  junc- 
tion from  1  to  2  there  is  a  rise  in  temperature ;  and  if,  on 
reversing  the  current,  there  is  a  fall  in  temperature,  it  is 
proved  that  there  is  a  different  of  potential,  i.  e.  an  E.  M.  F. 
acting  from  2  towards  1,  so  that  V\  is  naturally  greater 
than  F2.  If  the  heat  effect,  measured  in  ergs,  is  H ;  and  if 
the  intensity  of  the  current  is  i  and  it  flows  t  seconds  in 
producing  H,  the  E.  M.  F.,  E,  at  the  junction  is  given  by 
the  formula 

H=Eit, (1). 

i.  e.  E  =  HI  it       (la) 

This  E.  M.  F.  which  exists  at  the  junctions  of  any  two 
different  bodies  (conductors  or  dielectrics),  and  which  be- 
comes evident  whenever  two  bodies  are  brought  in  contact, 
is  called  the  "  Peltier  E.  M.  F."  (Its  effect  is  often  masked 
by  the  chemical  action  of  surrounding  media,  e.  g.  air,  upon 
the  two  substances.)  It  may  be  measured  as  just  shown  ;, 
and  exact  measurements  prove  that  it  changes  as  the  tem- 
perature does,  but  at  different  rates  for  different  pairs  of 
substances. 

It  may  also  be  proved  by  direct  experiment  that  in  a 
conductor,  different  points  of  which  are  at  different  tem- 
peratures, there  are  differences  of  potential.  These  are 


310  THEORY  OF  PHYSICS  [CH.  Ill 

called  "  Thomson  E.  M.  F.'s  " ;  and  in  some  metals  their 
directions  are  from  points  of  high  to  points  of  low  tem- 
peratures, and  in  others  are  in  the  opposite  direction. 

As  a  consequence  of  these  two  kinds  of  E.  M.  F.'s  in  a 
circuit  of  two  conductors,  whose  junctions  are  at  differ- 
ent temperatures,  there  are  currents  produced,  unless  the 
various  forces  balance  one  another.  If  the  temperature  of 
one  junction  is  kept  fixed,  and  that  of  the  other  varied,  it 
is  always  possible  to  find  such  a  temperature  for  it  that 
there  is  no  current.  This  second  temperature  is  called  the 
temperature  of  "inversion,"  corresponding  to  the  fixed 
temperature  of  the  other  junction  ;  for  if  the  varying 
temperature  is  now  changed  so  as  to  be  farther  away 
from  the  fixed  temperature,  the  current  is  reversed.  The 
energy  which  is  required  to  maintain  a  thermo-electric  cur- 
rent is  furnished,  as  was  said  before,  by  the  bodies  which 
keep  the  temperatures  of  the  junctions  fixed. 

By  having  a  series  of  thermo-electric  junctions,  that  is, 
by  joining  in  series  several  "  couples  "  of  two  conductors, 

and  by  keeping  the  alternate 
junctions  at  different  tempera- 
tures, it  is  possible  to  produce 
a  considerable  E.  M.  F.  The 
arrangement  is  shown  in  the 
figure ;  and  if  the  junctions 
on  the  side  A  are  at  even  a 
slightly  different  temperature 

from  those  on  the  side  B,  there  will  be  a  noticeable  E.  M.  F., 
which  can  produce  a  current.  Such  an  instrument  is 
called  a  "  thermopile,"  and  it  is  extremely  useful  in  de- 
tecting differences  in  temperatures. 

242.  Primary  Cells.  If  any  solid  dips  in  any  liquid, 
there  is,  of  course,  a  difference  of  potential  between  them  • 
and  this  is  found  to  be  especially  large  if  both  the  solid 
and  liquid  are  conductors.  The  value  of  the  difference  of 
potential  depends  upon  the  nature  of  the  two  substances  ; 


242] 


ELECTRIC   CONDUCTION 


311 


and,  if  two  solid  conductors  dip  in  the  same  liquid  conduc- 
tor, they  will  as  a  rule  be  at  different  potentials.  For  the 
liquid  conductor  has  the  same  potential  throughout,  and 
the  potentials  of  the  two  solid  conductors  differ  from  that 
of  the  liquid  by  different  amounts.  So,  if  the  solid  con- 
ductors are  now  joined  by  a  wire,  there  will  be  a  current 
produced  in  it  by  the  E.  M.  F.  at  its  two  ends,  because  this 
E.  M.  F.  is  maintained  by  the  differences  between  the  two 
solid  conductors  and  the  liquid.  Such  an  apparatus  is 
called  a  "  primary  cell." 

A  special  case  of  this  process  is  when  rods  of  zinc  and 
copper  dip  in  acidulated  water  (e.  g.  a  dilute  solution  of 
sulphuric  acid).  Both  metals  are  found 
to  have  potentials  lower  than  that  of  the 
liquid ;  but  that  of  the  copper  is  higher 
than  that  of  the  zinc,  and  so  the  E.  M.  F. 
produces  a  current  from  copper  to  zinc,  to 
acidulated  water,  to  copper,  etc.  It  is  ob- 
served, also,  by  experiment  that,  as  the 
current  flows,  the  zinc  rod  dissolves,  or 
wastes  away,  and  that  at  the  copper  rod 
hydrogen  gas  forms  and  bubbles  off  at  the  surface  of  the 
liquid. 

Before  the  zinc  and  copper  rods  are  joined  by  the  out- 
side wire,  the  relative  differences  in  potential  may  be  repre- 
sented by  the  diagram, 
in    which    vertical    dis- 
tances mean    differences 

Cu    in  potential.     Thus,  the 

zinc  rod  (Zn)  has  a  con- 
stant potential,  which  is 
shown  by  a  horizontal 
line ;  in  passing  from  the 

zinc  to  the  acidulated  water  there  is  a  marked  increase  in 
potential,  E\ ;  the  liquid  has  a  constant  potential  through- 
out, being  a  conductor;  then  in  passing  from  it  to  the 


169. 


LIQUID 


Zn 


FIG.  170. 


312 


THEORY  OF  PHYSICS 


[CH.  Ill 


copper  (Cu)  there  is  a  fall  of  potential  E2,  but  the  copper 
is  at  a  higher  potential  than  the  zinc,  since  E\  >  Ez. 

If  the  copper  and  zinc  rods  are  joined  by  a  copper  wire, 
a  current  is  thus  produced ;  and,  since  in  any  conductor  the 
current  flows  from  high  to  low  potential,  there  must  be  a 


Zn 


Cu  Cu.Zn 

FIG.  171. 


fall  of  potential  in  every  conductor  through  which  the  cur- 
rent is  passing,  as  is  shown  in  the  diagram.  Starting  from 
the  side  of  the  zinc  rod  next  the  liquid,  there  is  a  rise  in 
potential  to  the  liquid,  a  fall  through  the  liquid,  a  drop  in 
passing  from  the  liquid  to  the  copper,  a  fall  through  the 
copper  rod  and  wire,  a  slight  difference  at  the  junction  of 
the  copper  and  the  zinc,  a  fall  through  the  zinc  rod  at  a 
different  rate  from  the  fall  through  the  copper,  etc.  These 
facts  may  be  proved  by  direct  experiment. 

If  the  difference  of  potential  between  the  zinc  rod  and 
the  liquid  is  EI,  the  work  required  to  make  a  current  whose 
intensity  is  i  pass  for  t  seconds  from  the  zinc  into  the 
liquid  is  E\it.  Similarly,  if  the  difference  of  potential 
between  the  liquid  and  the  copper  is  Ez,  the  energy  set 
free,  as  the  current  passes  from  high  to  low  potential,  is 
E.2it\  that  is,  this  amount  of  electric  energy  is  transformed 
into  some  other  kind  of  energy.  In  a  similar  manner, 
energy  is  set  free  throughout  each  conductor  in  the  circuit, 
as  the  potential  falls  in  the  direction  of  the  current.  (The 
slight  Peltier  E.  M.  F.  at  the  junction  of  the  copper  and  zinc 
also  requires  or  furnishes  energy ;  but  this  amount  is  very 


242]  ELECTRIC  CONDUCTION  313 

minute.)  The  energy  set  free  by  the  current  as  it  passes  is 
spent  in  doing  work  at  the  junctions  or  in  the  conductors. 
As  the  current  passes  from  the  liquid  to  the  copper,  chemi- 
cal actions  take  place,  viz.  hydrogen  gas  is  formed,  and  the 
energy  JEzit.is  spent  in  doing  this  "chemical"  work. 
Through  the  conductors  there  is  a  fall  of  potential,  E\  —  E2 
(neglecting  the  Peltier  E.  M.  F.  at  the  Zn  — Cu  junction); 
and  the  energy  (E±  —  Ez)  i  t  is  spent  in  giving  heat-energy 
to  the  molecules  of  the  conductors.  Consequently,  in  gen- 
eral, their  temperature  is  raised.  Therefore,  the  electric 
energy  E\it  must  be  furnished  from  some  source  of  energy 
in  between  the  zinc  and  the  acidulated  water  \  E2it  is  spent 
in  chemical  work ;  (Ei  —  JS2)  i  t,  in  heat  effects.  At  the 
zinc  plate,  as  stated  before,  there  is  solution  of  the  zinc  in 
the  acidulated  water ;  and,  as  is  well  known,  when  zinc  dis- 
solves in  acid,  energy  is  liberated,  so  that  the  energy  E\  i  t, 
which  is  required  for  the  current,  is  furnished  by  the  mole- 
cules which  dissolve  from  the  zinc  into  the  acid.  (Some 
heat  energy  may  be  furnished  from  neighboring  bodies,  or 
may  be  given  up  to  them ;  but  these  amounts  are  here 
neglected.) 

If  1  gram  of  zinc  is  dissolved  in  a  vessel  containing  acidu- 
lated water,  hydrogen  is  evolved,  and  the  temperature  is 
raised.  By  measuring  the  rise  in  temperature  the  masses 
of  the  substances  and  their  specific  heats,  it  is  possible  to 
calculate  how  much  heat  energy  has  been  produced  when 
the  1  gram  of  zinc  dissolves  and  the  corresponding  amount 
of  hydrogen  is  produced.  Let  this  amount  of  energy  be  h. 
Then  if,  as  the  current  of  intensity  i  passes  for  t  seconds  in 
the  cell  which  has  been  described  above,  m  grams  of  zinc 
dissolve  and  a  corresponding  amount  of  hydrogen  is  pro- 
duced, the  energy  m  h  is  furnished  as  a  result  of  the  two 
chemical  actions  at  the  zinc  and  copper  poles.  Conse- 
quently, 

mh  =  E1it—Esit  =  (E1-E2)it      .     .     (2) 

So  that,  if  m,  h,  i,  t  are  known,  El  —  E2  may  be  calculated. 


314 


THEORY  OF  PHYSICS 


[CH.  Ill 


A  mechanical  analogy  may  make  the  action  of  a  pri- 
mary cell  clearer.  If  in  a  U-tube,  partly  filled  with  water,  a 
pump. or  paddle-wheel  is  inserted  at  the  bottom  and  worked 
by  some  external  power,  the  water  will 
be  raised  higher  in  one  arm  than  the 
other;  and,  if  the  two  arms  are  con- 
nected above  by  a  pipe,  the  water  may 
be  forced  over  from  one  arm  into  the 
other.  The  process  will  continue  in- 
definitely, as  long  as  energy  is  furnished 
the  pump  so  as  to  make  it  work.  The 
pump  forces  the  water  from  a  low  pres- 
sure on  one  side  to  a  high  one  on  the 
other,  just  as  the  chemical  action  at  the 
zinc  pole  raises  the  electricity  from  a 
low  potential  to  a  high  one. 

Such  a  primary  cell  as  the  one  de- 
scribed is  sometimes  called  a  "  single- 
fluid"  cell,  and  the  current  which  it 
furnishes  is  by  no  means  constant.  The  hydrogen  which  is 
formed  at  the  copper  pole  does  not  all  bubble  off ;  but  some 
remains  there,  changing  the  character  of  the  E.  M.  F. ;  some 
zinc  is  also  slowly  deposited  on  the  copper.  Again,  as  the 
positive  charges  are  carried  through  the  liquid,  they  are 
not  all  handed  on  to  the  copper  pole,  but  some  remain  near 
it,  thus  opposing  the  approach  of  other  positive  charges. 
This  last  fact  may  be  expressed  by  saying  that  there  is  an 
E.  M.  F.  in  the  opposite  direction  to  the  one  producing  the 
flow  through  the  liquid.  The  entire  phenomenon  is  called 
"  polarization,"  and  the  "  back  -  E.  M.  F."  is  called  the 
E.  M.  F.  of  polarization.  To  obviate  these  difficulties  certain 
cells  have  been  devised  which  contain  two  liquids  sepa- 
rated by  some  porous  partition,  thus  keeping  the  liquids 
apart,  but  permitting  the  current  to  pass.  In  these  cells 
polarization  may  be  largely  avoided,  and  so  a  fairly  con- 
stant current  may  be  produced.  If  the  current  is  too  great, 


FIG.  172. 


243] 


ELECTRIC  CONDUCTION 


315 


the  E.  M.  F.  of  any  cell  will  change.  The  best-known  cell 
of  a  two-fluid  type  is  the  Daniels.  In  this  the  two  fluids 
are  solutions  of  copper  sulphate  and 
sulphuric  acid  in  water;  and  the 
two  metals  are  zinc,  dipping  in  the 
acid,  and  copper,  dipping  in  the  cop- 
per sulphate.  Another  cell,  which 
is  more  complicated,  but  which  main- 
tains indefinitely  a  constant  E.  M.  F., 
provided  that  only  a  minute  current 
passes  through  it,  is  a  Clark  cell, 
which  is  now  used  in  laboratories  to 
give  a  standard  E.  M.  F. 

When  a  piece  of   pure  metal  is 
immersed   in   an  acid,  it   does   not 


FIG.  173. 


continue  to  dissolve ;  on  the  contrary,  the  process  ceases 
so  quickly  that  it  is  detected  with  difficulty.  •  But,  if 
there  is  any  metallic  impurity  ii\  the  metal,  the  process 
continues  until  the  impurity  is  dislodged.  The  explana- 
tion is  very  simple.  The  metallic  impurity,  the  metal 
itself,  and  the  acid  form  a  primary  cell,  like  the  copper- 
zinc-acid  cell ;  and,  as  the  current  passes  from  the  metal 
into  the  acid,  the  metal  is  dissolved,  just  as  the  zinc  was. 
Ordinary  commercial  zinc  rods  are  quite  impure;  so  that, 
when  they  are  dipped  in  dilute  acids,  some  zinc  dissolves 
without  producing  any  current  outside.  This  is  called 
"  local  action  ; "  and  it  may  be  largely  prevented  by  amal- 
gamating the  surface  of  the  zinc  with  mercury,  so  as  to 
render  the  surface  approximately  uniform. 

243.  Nature  of  Conductors.  All  metals,  in  the  solid  and 
liquid  conditions,  are  conductors ;  but  only  a  limited  num- 
ber of  other  liquids  are;  and  gases  conduct  only  under 
definite  conditions.  There  are  certain  complex  conductors 
such  that,  when  a  current  is  passed  through  them,  some  of 
their  constituents  are  liberated  at  the  place  where  the  cur- 
rent enters,  and  other  constituents  at  the  place  where  the 


316  THEOEY  OF  PHYSICS  [CH.  Ill 

current  leaves.  Thus,  when  a  current  is  passed  through  a 
solution  of  sulphuric  acid  in  water,  hydrogen  and  oxygen 
are  liberated  at  the  places  where  the  current  leaves  and 
enters,  respectively.  Again,  if  a  current  is  passed  through 
a  solution  of  silver  nitrate  in  water,  silver  and  oxygen  are 
liberated.  Such  conductors  are  called  "  electrolytes ; "  and 
the  process  of  the  conduction  through  them  is  called 
"  electrolysis." 

244.  Electrolysis.  Some  solids  are  electrolytes,  such  as 
silver  iodide.  With  the  exception  of  fused  metals,  all 
liquid  conductors  are  electrolytes ;  and,  when  a  gas  is  made 
conducting,  it  becomes  an  electrolyte.  The  metal  con- 
ductors, sometimes  called  "  electrodes  "  or  "  poles,"  by  which 
the  current  enters  and  leaves  a  liquid  or  gaseous  conductor, 
are  called  "  anode  "  and  "  kathode,"  respectively. 

The  laws  of  conduction  through  electrolytes  have  been 
investigated  by  Faraday,  and  may  be  thus  expressed : 

1.  The  mass  of  the  substance  liberated  in  a  given  time 
at  either  kathode  or  anode  is  directly  proportional  to  the 
quantity  of  electricity  which  has  passed,  i.  e.  to  the  prod- 
uct of  the   intensity  of   the  current  and  the  number  of 
seconds,  i  t. 

2.  If  the  same  current  is  passed  through  several  electro- 
lytes in  series,  i.  e.  if  several  substances  are  being  liberated 
at  the  same  time,  the  masses  of  the  substances  liberated  are 
directly  proportional  to  their  "chemical  equivalents."     By 
definition,  the  chemical  equivalent  of  any  element  is  its 
"  atomic  weight "  divided  by  its  "  valence."     And  by  "  val- 
ence" is    meant   the  number  of  hydrogen    atoms    which 
combine  with  one  atom  of  the  element  to  form  a  stable 
compound ;  or,  what  is  the  same  thing,  twice  the  number 
of  oxygen  atoms  which  combine  with  one  atom    of  the 
element   to  form   a   compound.      In    a   similar  way,  the 
valence  of  any  group  of  atoms  or  "  radical "  may  be  defined 
as  the  number  of  hydrogen  atoms  or  twice  the  number  of 
oxygen  atoms  whic'n  combine  with  it  to  form  a  molecule. 


245] 


ELECTRIC  CONDUCTION 


317 


Thus,  the  valence  of  S  04  is  2,  because  there  is  a  molecule, 
H2  S  04,  sulphuric  acid. 

The  following  table  gives  the  u  atomic  weight,"  valency, 
and  chemical  equivalent  of  several  elements :  — 

TABLE   XIII 


Atomic 
Weight. 

Valeucy. 

Chemical 
Equivalent. 

Chlorine  ....... 

35.37 

1 

35.37 

Copper  (Cupric)    .... 

63.18 
1.00 

2 
1 

31.59 
1.00 

55.88 

3 

18.63 

Lead   .          .               ... 

206.39 

2 

103.20 

Oxvaren 

15.96 

2 

7.98 

39.03 

1 

39.03 

Silver  

107.66 

1 

107.66 

23.00 

1 

23.00 

Zinc    

64.88 

2 

32.44 

Faraday's  first  law  furnishes  at  once  a  method  by  means 
of  which  the  intensities  of  two  currents  may  be  compared. 
If  the  mass  ml  is  liberated  by  a  current  whose  intensity 
is  i\  flowing  for  ti  seconds,  and  if  m2  is  liberated  in  the 
same  electrolyte  when  a  current  of  intensity  i2  flows  for 
tz  seconds, 

77i i  :  ra2  =  ii  ti  :  i2  U, 
or  ii  :  iz  =  miU  :  ?nzti (3) 

And  m1?  w2,  h,  and  tz  can  be  easily  measured.  This  prin- 
ciple is  made  use  of  in  "gas  voltameters,"  where  gases 
are  liberated ;  and  in  "  copper  and  silver  voltameters,"  in 
which  copper  and  silver,  respectively,  are  deposited  at  the 
kathodes.  An  illustration  of  a  common  type  of  gas  vol- 
tameter is  given  on  the  following  page. 

245.   From  a  study  of  electrolysis  and  its  laws  as  dis- 


318 


THEORY  OF  PHYSICS 


[CH.  Ill 


covered  by  Faraday,  a  great  deal  may  be  learned  as  to  the 
nature  of  conduction.  The  mechanism  of  the  conduction 
is  the  passage  of  a  series  of  positive 
charges  in  one  direction  and  of  nega- 
tive ones  in  the  other,  these  charges 
being  carried  on  portions  of  matter ; 
because  there  is  no  steady  current 
unless  portions  of  matter  are  liber- 
ated at  the  poles,  and  the  amount  of 
the  current  is  directly  proportional 
to  the  amounts  of  matter  set  free. 
The  name  given  these  portions  which 
carry  the  charges  is  "i°ns."  By  ex- 
amining at  which  pole,  anode  or  ka- 
thode, a  given  substance  is  liberated, 
the  kind  of  electric  charge  on  the 
ions  may  be  determined.  If  a  sub- 
stance is  liberated  at  the  kathode, 
i.  e.  where  the  current  leaves,  its  ions 
must  have  been  positively  charged  ; 
and,  if  liberated  at  the  anode,  nega- 
tively charged. 
Experiments  seem  to  prove  that  in  every  case  of  electrol- 
ysis the  ions  are  dissociated  portions  of  molecules.  Thus, 
no  gas  conducts  until  some  of  the  molecules  are  disso- 
ciated, —  a  state  of  affairs  which  can  be  proved  to  exist  by 
other  methods.  Again,  with  the  exception  of  fused  sub- 
stances, no  pure  liquid  conducts,  some  salt  or  acid  must 
be  added  to  it ;  and  there  is  a  great  deal  of  independent 
evidence  leading  to  the  belief  that  no  such  liquid  is  a  con- 
ductor unless  the  salt  or  acid  in  solution  in  it  is,  at  least 
partially,  dissociated. 

Faraday's  second  law  fixes  the  relative  charges  carried 
by  an  ion  of  any  form  of  matter.  First,  the  charge  carried 
on  any  one  ion  of  the  same  substance  is  always  the  same, 
no  matter  in  what  electrolyte  the  ion  exists.  This  is 


FIG.  174. 


245]  ELECTRIC  CONDUCTION  319 

proved  by  the  following  fact  :  if  the  same  current  is  passed 
through  two  electrolytes  in  series,  which  liberate  at  their 
two  similar  poles,  e.  g.  kathodes,  the  same  substance  (such 
as  solutions  of  sulphuric  acid  and  hydrochloric  acid  in 
water,  in  both  of  which  hydrogen  is  liberated  at  the  ka- 
thodes), the  masses  there  set  free  are  the  same,  the  chemical 
equivalent  of  any  one  substance  being  always  the  same  ; 
but  the  quantities  of  electricity  carried  through  each  electro- 
lyte are  the  same,  and  they  are  carried  on  the  same  number 
of  ions  in  each,  since  the  masses  are  equal  ;  consequently  the 
charge  carried  on  an  ion  in  the  one  electrolyte  is  the  same 
as  that  carried  on  the  same  ion  in  the  other  electrolyte. 
There  is,  then,  always  associated  with  an  ion  of  any  sub- 
stance a  definite  charge  which  cannot  be.  varied;  it  is  just 
as  much  a  property  of  the  ion  as  its  mass. 

Second,  the  charges  carried  on  ions  of  different  sub- 
stances are  in  the  ratio  of  their  valences.  This  is  proved  by 
the  following  deduction  from  Faraday's  laws  :  let  the  same 
current  pass  through  two  electrolytes  in  series,  so  that  the 
same  quantity  of  electricity  is  being  carried  by  two  sets  of 
ions.  A  certain  number  of  ions  in  each  set  are  liberated  in 
a  certain  time  ;  call  these  numbers  n\  and  nz  for  the  two 
sets  ;  and  call  the  electric  charges  carried  on  each  ion  of 
the  two  sets  el  and  e2.  Then 

7i!  d  =  n2  e2, 

since  the  same  total  charges  are  being  carried.  Further, 
if  mi  and  mz  are  the  masses  of  the  two  substances  set 
free,  and  w±  and  w2  the  actual  masses  of  an  ion  in  each 
of  the  two  sets, 

77?  i  =  tti  Wi, 


that  is,  mi  :  mz 

But  by  Faraday's  second  law,  writing  Ci  and  c2  for  the 
chemical  equivalents  of  the  two  ions, 

m\  :  m2  =  c{  :  c2   ......     (4) 


320  THEORY   OF  PHYSICS  [cH.  Ill 

Substituting  for  mx  /  m2  its  value,  this  becomes 


But  w1/w2  =  ratio  of  "atomic  weights"  by  definition  of 
"  atomic  weight  ;  "  and  hence,  since  c  is  the  ratio  of  atomic 
weight  to  valence,  a  /  e2  =  ratio  of  the  valences  of  the  two 
substances.  So,  if  a  hydrogen  ion  carries  a  charge  e,  any 
other  ion,  whose  valence  is  v,  must  carry  a  charge  v  e. 

This  fact  follows  at  once  if  ions  are  regarded  as  disso- 
ciated parts  of  molecules.  For  a  molecule  is  uncharged 
or  neutral  ;  arid  so,  if  it  breaks  up  into  ions,  some  will  be 
+,  the  others  —  ;  but  the  amount  of  each  charge  must  be 
the  same.  Thus,  if  a  molecule  of  hydrochloric  acid  breaks 
up  into  an  hydrogen  ion  and  a  chlorine  ion, 

HC1  =  H  +  Cl, 

the  charge  on  the  hydrogen  ion  must  be  equal  but  opposite 
to  that  on  the  chlorine  ion.  If  a  molecule  of  zinc  chloride 
breaks  up  into  a  zinc  ion  and  two  chlorine  ions 

ZnCl2  =  Zn  +  Cl  +  Cl, 

the  charge  on  the  zinc  ion  must  be  twice  as  great  as  that 
on  a  chlorine  ion,  and  therefore  twice  as  great  as  that  on  a 
hydrogen  ion.  But  the  valence  of  zinc  is  two.  All  other 
ions  may  be  considered  in  the  same  way  ;  and  the  above 
law  is  a  natural  consequence. 

If  the  electrolyte  is  a  liquid,  the  same  ion  is  always 
charged  with  the  same  kind  of  a  charge  ;  thus  a  hydrogen 
or  a  metal  ion  of  any  kind  is  always  charged  positively, 
an  oxygen  or  chlorine  ion  always  negatively.  This  is  not 
true  if  the  electrolyte  is  a  gas  ;  for  there  a  hydrogen  ion  is 
under  certain  conditions  positive,  under  others  negative  ; 
and  similarly  with  an  oxygen  or  chlorine  ion.  The 
amount  of  the  charge  is,  however,  always  the  same  for 
any  one  ion. 

246.  It  is  a  matter  of  easy  experiment  to  determine 
how  many  grams  of  any  substance  are  liberated  from  an 


246]  ELECTRIC   CONDUCTION  321 

electrolyte  when  a  certain  current  passes ;  and,  if  this  is 
known  for  any  one  substance,  the  amounts  of  any  other 
substance  which  would  be  liberated  by  the  same  current 
may  be  calculated  from  Faraday's  second  law.  The  amounts 
which  would  be  liberated  by  a  different  current  may  be 
calculated  by  Faraday's  first  law.  The  quantity  of  elec- 
tricity, that  is,  the  intensity  of  the  current  multiplied  by 
the  number  of  seconds  (i  t),  would  be  naturally  calculated 
in  terms  of  the  electrostatic  unit  (see  Art.  218) ;  but  it  is 
practically  impossible  to  measure  currents  in  terms  of  this 
unit.  So  another  unit  is  ordinarily  used  for  measuring 
currents ;  and,  if  its  value  is  determined  once  for  all  in 
terms  of  the  electrostatic  unit,  the  current  may  be  ex- 
pressed in  terms  of  it  also,  if  it  is  desired.  This  new 
unit  of  quantity  for  measuring  currents  will  be  defined 
later  (see  Art.  277)  ;  and  it  is  called  the  "  electromagnetic 
unit."  The  number  of  grams  of  any  substance  liberated 
when  1  electromagnetic  unit  passes  (i.  e.  the  number  of 
grams  of  a  definite  kind  of  ions,  which  carry  a  charge 
equal  to  1  unit)  is  known  as  the  "  electro-chemical  equiva- 
lent" of  that  substance.  Some  of  these  values  for  differ- 
ent elements  are  given  in  the  following  table :  — 

TABLE   XIV 
ELECTRO-CHEMICAL  EQUIVALENTS 


Chlorine       .     . 
Copper  (Cupric) 
Hydrogen    .     . 

0.0003675 
0.003261 
0.00010352 

Iron  (Ferric) 
Oxygen    .     .     . 
Silver  .... 

0.001932 
0.000828 
0.011180 

It  should  be  observed  that,  if  a  certain  number  of  units  of 
quantity  liberate  a  number  of  grams  of  a  substance  equal 
to  its  chemical  equivalent,  that  same  quantity  will  liber- 
ate a  number  of  grams  of  any  other  substance  equal  to  its 
chemical  equivalent.  For,  by  Faraday's  second  law, 

mi  :  mz  =  Ci  :  c2. 
11 


322  THEORY  OF  PHYSICS  [CH.  Ill 

And,  if  mi  =  cl}  m2  must  equal  c2.  It  may  be  easily  cal- 
culated from  the  table  just  given  that  9654  units  will  set 
free  an  amount  of  any  substance  whose  mass  numerically 
equals  its  chemical  equivalent.  (The  chemical  equivalent 
of  hydrogen  is  1 ;  hence  1  gram  of  hydrogen  ions  carry 
9654  electromagnetic  units.) 

247.  Dissociation  Theory  of  Electrolytes.  As  already  stated, 
the  laws  of  electrolysis  may  all  be  explained  if  the  ions  are 
dissociated  portions  of  molecules,  each  one  carrying  a  defi- 
nite charge  proportional  to  its  valence.  This  theory  will 
now  be  explained  more  in  detail. 

In  gases  the  explanation  is  extremely  simple.  It  is 
known  that  gases  can  be  dissociated  in  several  ways,  by 
raising  the  temperature  sufficiently,  by  producing  so  great 
an  electric  strain  that  a  spark  passes,  etc.  ;  and  experiments 
prove  that,  if  there  is  any  dissociation  in  a  gas,  the  gas 
can  conduct  a  current,  and,  if  there  is  no  dissociation,  it 
cannot. 

In  the  case  of  liquid  electrolytes,  the  explanation  is  more 
complicated.  It  must  first  be  noted  that,  with  the  excep- 
tion of  certain  fused  solids,  no  liquid  can  conduct  a  current 
unless  there  is  dissolved  in  it  some  salt  or  acid.  Further, 
only  particular  kinds  of  liquid  solutions  can  conduct ;  and 
in  all  these,  with  no  exception,  there  are  independent  rea- 
sons for  believing  that  the  salt  or  acid  is  dissociated.  The 
theory  is  that,  when  a  salt  or  acid  is  dissolved  in  certain 
liquids,  there  is  partial  dissociation.  This  is  particularly 
true  of  ordinary  salt  or  acid  solutions  in  water,  but  is  not 
true  of  most  organic  solutions,  such  as  sugar  in  water  or 
salt  in  benzine.  If,  for  example,  a  gram  of  common  salt, 
sodium  chloride  (NaCl),  is  put  in  water,  a  definite  per- 
centage of  the  salt  will  dissociate  into  sodium  and  chlorine 
ions.  These  ions  are  not  molecules,  but  are  charged  atoms 
in  this  case ;  and  so  it  would  not  be  expected  that  the 
usual  properties  of  solid  sodium  or  chlorine  gas  would  be 
shown.  The  sodium  ion  is  positively  charged ;  the  chlo- 


247]  ELECTRIC  CONDUCTION  323 

rine  ion  negatively,  as  is  proved  by  passing  a  current 
through  the  liquid,  and  seeing  which  substance  comes  with 
the  current  to  the  kathode,  and  which  against  the  current 
to  the  anode.  In  this  solution  of  common,  salt  in  water 
the  percentage  of  molecules  dissociated  is  supposed  to  re- 
main constant  (unless  temperature  or  other  conditions 
change) ;  but  the  process  of  dissociation  and  combination 
is  supposed  to  go  on  continually,  so  that  any  individual  ion 
may  exist  as  such  for  a  while,  and  may  then  combine  with 
an  ion  of  the  other  kind  to  form  a  molecule.  This  process 
is  supposed  to  be  going  on  in  any  solution  of  common  salt  in 
water.  If,  now,  a  difference  of  potential  is  applied  at* two 
points  in  the  solution  by  two  wires  joined  to  a  primary  cell 
(or  any  other  source  of  E.  M.  F.),  there  will  be  an  electric 
force  urging  the  positively  charged  ions  towards  the  ka- 
thode during  those  intervals  of  time  in  which  these  ions  are 
dissociated ;  similarly,  the  negatively  charged  ions  will  be 
urged  in  the  opposite  direction,  towards  the  anode.  Any 
individual  ion  will  be  given  a  velocity  in  one  direction,  may 
then  meet  another  ion  of  the  other  kind,  with  which  it 
combines,  and  will  then  receive  no  further  acceleration 
until  it  becomes  free  again.  It  must  be  noticed  that,  on 
this  theory,  the  dissociation  is  not  produced  by  the  E.  M.  F., 
or  the  current,  but  takes  place  as  a  part  of  the  process  of 
solution. 

In  any  actual  case  it  is  exceedingly  difficult  to  determine 
exactly  what  the  ions  are.  If  pure  water  is  dissociated  at 
all,  it  must  be  to  an  amount  almost  too  minute  to  measure. 
Any  substance  which  goes  into  solution  and  dissociates 
forms  ions  ;  and  these  are  carried  to  the  anode  and  kathode. 
The  actions  at  these  places,  however,  are  complicated. 
Some  ions  combine  with  each  other  to  form  molecules ; 
others  react  on  the  liquid,  and  the  products  of  the  reaction 
form  molecules  and  are  liberated.  Of  course  no  substance 
can  be  liberated  unless  it  exists  in  the  form  of  molecules. 
Consider  several  cases. 


324  THEORY  OF  PHYSICS  [CH.  Ill 

1.  Dilute  aqueous  solution  of  sulphuric  acid,  H2S04,  with 
platinum  wires  as  anode  and  kathode.     The  ions  are  sup- 
posed to  be 

H,  H,  S04. 

The  H  ions  go  to  the  kathode,  give  their  charges  to  the 
metal  conductor  there,  combine  to  form  molecules,  H2,  and 
so  bubble  off  as  a  gas.  The  S04  ions  go  to  the  anode,  give 
up  their  charges  to  the  metal  conductor,  but  cannot  escape, 
because  of  themselves  they  cannot  form  molecules,  neither 
can  they  combine  with  the  platinum  wire ;  so  there  is  a  re- 
action with  the  water, 

2  H20  +  2  S04  =  2  H2S04  +  02, 

and  the  oxygen  gas  bubbles  off. 

2.  A  solution  of  common  salt,  NaCl,  in  water,  with  plati- 
num wires  for  anode  and  kathode.     The  ions  are  supposed 
to  be  Na,  Cl.     The  Na  ions  go  to  the  kathode  ;  and,  in 
general,  there  is  a  reaction  with  the  water. 

2  H20  +  2  Na  =  2  NaOH  +  H2 ; 

so  hydrogen  gas  escapes.  Chlorine  gas  escapes  at  the 
anode. 

3.  A  solution  of  copper  sulphate,  CuS04  in  water.     The 
ions  are  Cu,  S6~4.     The  Cu  ions  go  to  the  kathode,  where 
they  will  in  general  be  deposited  as  metallic  molecules. 
The  S04  ions  will  go  to  the  anode ;  and,  if  this  is  a  platinum 
wire,  oxygen  gas  will  be  evolved  as  in  case  1.     But,  if  the 
anode  is  a  copper  plate,  the  S04  can  react  on  the  copper, 
making  some  of  it  go  into  solution.     So  that,  in  this  last 
case,  copper  is  dissolved  from  the  anode  and  deposited  on 
the  kathode.     This  is  the  principle  of   "copper-plating." 
Silver-plating  may  be  done  in  a  similar  manner  by  using  a 
silver  solution  which  is  a  conductor  and  a  silver  plate  as 
an  anode. 

248.   As  stated  above  (Art.  242),  when  a  solid  conduc- 
tor dips  in  a  liquid  one,  there  is  always  a  differeace  of 


248]  ELECTRIC  CONDUCTION  325 

potential  between  the  two.  This  fact  may  be  explained 
at  once  on  the  dissociation  theory  of  electrolytes.  The 
simplest  case  is  when  a  metal  dips  into  a  solution  of  one 
of  its  own  salts,  e.  g.  zinc  into  a  solution  of  zinc  chloride 
(ZnCL).  When  the  solution  of  ZnCl2  is  made,  it  is  sup- 
posed that  some  of  the  molecules  dissociate,  thus  giving 
Zn  and  Cl  ions.  If  now  a  rod  of  zinc  dips  in  the  solution, 
some  of  the  zinc  in  general  dissolves  (unless  the  solution  is 
"  supersaturated  ")  ;  and,  since  experiments  show  that  in  a 
liquid  solution  of  any  metallic  salt,  the  metal  ion  is  always 
positively  charged,  the  zinc  ion  which  leaves  the  rod  and 
enters  the  solution  has  a  positive  charge.  But  there  must 
be  an  equal  negative  charge  left  on  the  zinc  rod;  and 
owing  to  the  action  of  positive  and  negative  charges,  the 
positive  charged  zinc  ions  in  the  solution  will  form  a  layer 
around  the  negatively  charged  zinc  rod.  Consequently 
there  will  be  a  difference  of  potential  between  these  two 
layers,  the  zinc  rod  being  lower  than  the  solution.  As  the 
process  continues,  the  electric  energy  of  these  two  layers 
increases  rapidly;  and,  when  any  further  solution  of  the 
zinc  rod  would  produce  an  increase  in  the  entire  potential 
energy  of  the  various  changes,  the  process  stops.  This  is 
the  reason  why  a  pure  metal  cannot  perceptibly  dissolve 
in  an  acid ;  an  electric  current  is  required.  Further,  since 
a  metal  ion  leaves  the  metal  carrying  with  it  a  positive 
charge,  it  is  seen  why  in  a  circuit  carrying  an  electric  cur- 
rent, that  metal  dissolves  at  whose  surface  the  current 
flows  into  the  electrolyte,  if  there  is  any  chemical  action 
between  the  solid  and  liquid ;  while  at  the  other  metal 
pole,  the  kathode,  a  metal  may  be  deposited. 

On  the  dissociation  theory,  no  charges  of  any  kind  can 
exist  except  on  portions  of  molecules,  i.  e.  ions.  A  charged 
conductor  has,  then,  on  its  surface  a  great  number  of  dis- 
sociated molecules;  and  the  process  of  charging  a  body 
consists  in  causing  dissociation  of  its  surface  molecules 
into  positive  and  negative  ions,  and  then  making  the 


326  THEORY  OF  PHYSICS  [CH.  Ill 

charges  all  the  same,  either  by  removing  those  ions  which 
are  charged  the  opposite  way  or  by  reversing  their  charges. 

249.  Process  of  Conduction.      The  process  of  conduction 
in  an  electrolyte  consists,  then,  in  the  handing  on  of  the 
positive  charges  in  one  direction,  and  of  the  negative  ones 
in   the  other,  both  charges  being  carried   by  portions  of 
matter  called  ions.     This  is  the  mode  of  action  in  most 
liquid  and  gaseous  conductors.    In  the  case  of  solid  metal 
conductors  the  process  may  be  just  the  same,  there  being 
no  reason  for  believing  it  different  in  principle.     Of  course, 
the  molecules  of  a  solid  cannot  travel  far,  —  they  can  simply 
oscillate  about  positions  of  equilibrium ;  and,  again,  only  a 
very  small  proportion  of  the  molecules  need  take  part  in 
the  conduction.     In  all  cases,  though,  the  process  is  essen- 
tially discontinuous. 

250.  Discharge  through   Gases.     When   an   electric   dis- 
charge, or  spark,   passes  through  a  gas,  there  are  many 
most  interesting  and  important  phenomena.     The  simplest 
illustration  of  such  a  discharge  is  given  by  a  flash  of  light- 
ning.    The  methods  commonly  used  for  producing  a  dis- 
charge are  to  cause  sparks  to  pass  by  means  of  an  electric 
machine,  such  as  a  Voss  or  a  Wimshurst  one,  or  by  means  of 
a  so-called  induction-coil  (see  Art.  289).     The  gas  through 
which  the  discharge  takes  place  is  rendered  luminous  by  its 
passage;  and  the  light  observed  is   characteristic  of   the 
vapor  or  gas,  and  also  of  the  metal  poles  between  which 
the  spark  passes,  because  in  the  act  of  discharge  some  of 
the  metal  molecules  are  torn  off  from  the  terminals  of  the 
machine  or  coil. 

The  nature  of  the  discharge  depends  largely  upon  the 
pressure  of  the  gas.  If  it  is  at  ordinary  pressures,  the  gen- 
eral appearance  is  like  that  of  the  lightning;  but,  if  the 
gas  is  enclosed  in  a  vessel  from  which  some  may  be  ex- 
hausted by  means  of  an  air-pump,  it  is  found  that  at  low 
pressures  the  character  of  the  discharge  entirely  changes. 
The  method  of  observation  generally  adopted  is  to  place 


250]  ELECTRIC   CONDUCTION  327 

the  gas  in  a  glass  bulb,  into  which  enter  metal  wires  ;  and 
to  study  the  effect  of  passing  discharges  between  these 
metal  electrodes,  at  different  pressures  of  the  gas. 

If  the  pressure  of  the  gas  is  in  the  neighborhood  of 
1  mm.  of  mercury,  the  typical  discharge  is  like  that  shown 
in  the  diagram.  At  the  anode  there  is  a  series  of  strati- 
fications or  "  strise,"  generally  beautifully  colored.  While 
at  the  kathode  there  is  a  dark  space  except  immediately  on 
the  surface  of  the  metal  pole.  These  two  phenomena  are 
entirely  distinct  and  independent  of  each  other.  It  has 
been  recently  shown  that  the  existence  of  strise  in  a  dis- 


FIG.  175. 

charge-tube  is  probably  positive  proof  of  the  presence  of  an 
impurity  in  the  gas.  There  is  also  another  phenomenon  at 
the  kathode,  which  deserves  notice,  and  which  is  best  ob- 
served if  the  vacuum  is  such  that  the  pressure  of  the  gas  is 
only  a  fraction  of  a  millimetre  of  mercury,  i.  e.  if  the  gas 
forms  the  "Fourth  State  of  Matter"  (see  Art.  210).  In  a 
gas  so  rarefied,  the  phenomena  at  the  anode  disappear,  the 
dark  space  from  the  kathode  fills  the  tube;  but  now  it 
may  be  plainly  observed  that  there  is  a  faint  radiation  from 
the  face  of  the  metal  forming  the  kathode,  and  that  it 
takes  place  in  straight  lines.  This  may  be  shown  by  the 
faint  luminescence  of  the  gas  through  which  the  rays  pass, 
and  by  the  brilliant  colored  effects  produced  where  the 
rays  strike  the  glass  walls  of  the  tube  or  other  substances 
which  emit  light  under  these  conditions.  By  putting 
obstacles  between  the  kathode  and  the  walls  of  the  tube,  it 
may  be  shown  that  the  rays  pass  in  straight  lines,  because 
such  sharp  shadows  are  cast.  These  rays  are  called  the 
"  kathode  rays  ;  "  and  they  have  been  proved  to  consist  of 
streams  of  portions  of  matter,  negatively  charged,  moving 


328  THEORY   OF  PHYSICS  [CH.  Ill 

away  from  the  kathode  with  a  great  velocity.  These  rays 
can  pass  through  many  obstacles  if  they  are  thin  enough, 
such  as  aluminium  foil ;  they  can  affect  a  photographic 
plate  ;  they  are  deflected  by  a  magnet  (like  any  electric 
current). 

Besides  these  two  radiations  produced  in  a  highly  rare- 
fied gas,  viz.  light  waves  and  "  kathode  rays,"  there  is  an- 
other which  is  not  yet  understood.  When  the  discharge 
is  taking  place  through  the  tube,  it  may  be  proved  that, 
proceeding  out  from  some  points  inside,  through  the  glass, 
there  is  a  distinctly  new  radiation.  This  was  discovered 
by  Professor  Kontgen  of  Wiirzburg,  Germany ;  and  many 
of  its  properties  are  now  known.  The  rays  proceed  in 
straight  lines ;  they  are  not  deflected  by  a  magnet ;  they 
make  many  substances  emit  light  when  they  fall  upon 
them ;  they  can  pass  through  many  obstacles  which  are 
opaque  to  ordinary  light  waves ;  they  affect  photographic 
plates  ;  they  change  any  insulating  medium  or  dielectric, 
such  as  air,  paraffin,  etc.,  into  a  conductor  when  they  pass 
through  it. 


CHAPTEE  IV 

PROPERTIES   OF   STEADY   ELECTRIC   CURRENTS 

WHEN  an  electric  current  is  produced  by  applying  a 
constant  difference  of  potential,  or  E.  M.  F.,  to  a  conductor 
or  series  of  conductors,  certain  effects  are  noticed  in  the 
conductors,  and  these  take  place  according  to  certain  laws. 
Such  a  current  is  called  a  "  steady  "  current. 

251.  Uniformity  of  Current.     It  may  be  proved  by  most 
careful  experiments  that  the  intensity  of  a  steady  current 
is  the  same  in  all  parts  of  the  circuit,  no  matter  what  the 
conductor  is, — liquid,  gas,  or  solid.     And,  if  the  circuit 
branches   at  any  point,  so  that  two  or  more  conductors 
carry  the  current  away  from  that  point,  the  sum  of  the 
intensities  in  these  conductors  exactly  equals  the  inten- 
sity of  the  current  in  the  conductor  by  which  the  current 
is  brought  to  the  point  of  branching.     If  this  law  were  not 
true,  there  would  be  an  increasing  or  decreasing  statical 
charge  at  certain  points  in  the  circuit ;  and  this  would 
show  that  the  current  was  not  steady.     Further,  it  may  be 
proved  directly  by  showing  that  the  magnetic  effect   of 
the  current  (see  Art.  253)  is  the  same  at  all  points  of  the 
circuit. 

252.  Heating  Effect.      The   current  'always   flows   from 
high  potential  to  low ;  and  in  so  doing  energy  is  given  the 
minute  portions  of  matter  which  make  up  the  conductor ; 
electric   energy   disappears,  being   transformed   into   heat 
energy.     Stated  in  other  words,  work  is  required  to  make 
a  current  pass   along  a  conductor ;  and,  this  work  being 
done  against  forces  connected  with  the  smallest  portions 


330  THEORY  OF  PHYSICS  [CH.  IV 

of  the  conductor,  heat  effects  are  produced.  (Notice  the 
analogy  to  water  flowing  from  a  point  of  high  pressure  to 
one  of  low  through  a  pipe  where  there  is  a  good  deal  of 
friction.)  The  fact  that  the  temperature  of  a  conductor 
rises  when  a  current  passes  in  it,  is  familiar  to  every  one 
hy  a  great  many  illustrations,  such  as  the  ordinary  glow- 
lamp,  the  arc-lamp,  the  burning  out  of  dynamos  and  mo- 
tors, the  violent  sparks  produced  wherever  there  is  a  bad 
contact,  etc. 

The  work  done  in  any  particular  portion  of  the  conduct- 
ing circuit  may  be  easily  calculated.     Let  A  and  B  be  two 

points  on  any  conductor 
forming  part  of  the  cir- 
cuit through  which  a  cur- 

Fio  176  rent  of  intensity,  i,  flows. 

Let  the  current  flow  from 
A  to  B,  and  let  E  be  the  difference  of  potential  between 
them.  Therefore,  in  t  seconds  a  quantity  i  t  passes,  be- 
cause i  is  the  quantity  that  passes  in  1  second.  But  E  is 
the  amount  of  work  done  ly  a  unit  positive  quantity  of 
electricity  in  passing  from  A  to  B,  or  by  a  unit  negative 
quantity  in  passing  from  B  to  A.  So  in  t  seconds  the 
work  done  by  the  current  against  these  minute  forces  of 
the  conductor  is  Eit.  This  is  generally  called  the  heat- 
ing effect,  and  may  be  written 


(i) 


H  is,  of  course,  measured  in  ergs.  The  rise  in  temperature 
may  also  be  calculated.  If  m  is  the  mass  of  the  conductor 
between  A  and  B  ;  c,  its  specific  heat  ;  T,  the  rise  in  tem- 
perature ;  J,  the  mechanical  equivalent  of  heat  (see  Art. 
212),  4.2  x  107;  then 

H  =  mcTJ=Eit; 
T=^-T  (2) 


253]     PROPERTIES  OF   STEADY  ELECTRIC  CURRENTS    331 


253.  Magnetic  Effect.  If  a  magnetic  needle  is  sus- 
pended, free  to  turn,  near  a  current,  the  magnet  is  affected. 
In  particular,  if  the  current  is  horizontal,  and  the  mag- 
netic needle  is  suspended  over  it  and  parallel  to  it,  the 
magnet  is  deflected  so  as  to  stand  as  nearly  as  it  can  at 
right  angles  to  the  current.  The  fundamental  property 
of  a  magnetic  needle  is,  of  course,  that  it  tends  to  point  in 
a  direction  which  is  nearly  north  and  south ;  and  that  end 
which  tends  to  be  towards  the  north  is  called  the  "  north 
pole  "  of  the  magnet ;  and  the  other  end,  the  "  south  pole." 
It  may  be  proved  by  experiment  that  the  action  of  a  cur- 
rent on  a  magnet  may  be  thus  described :  the  north  pole 
of  a  magnetic  needle  tends  to  move  around  a  current  in 
such  a  direction  that  there  is  the  same  connection  between 
the  direction  of  the  current  and  this  direction  of  the  motion 
of  the  north  pole  of  the  magnet  as  there  is  between  the 
direction  in  which  an  ordinary  right-handed  screw  ad- 
vances into  a  board  and  the  direction  in  which  it  must  be 
turned  in  order  to  make  it  enter.  This  law  of  direction 
is  sometimes  called  "the  right-handed  screw  law."  The 
south  pole  of  the  mag- 
netic needle  tends  to  move 
in  an  opposite  direction. 
Consequently,  a  magnetic 
needle  will  tend  to  place 
itself  at  right  angles  to 
the  current.  In  particu- 
lar, if  a  magnet  is  sus- 
pended inside  a  coil  of 
wire,  parallel  to  the  coils, 
it  will  be  deflected  when 

there  is  a  current  through  the  wire.  Let  the  direction  of 
the  current  and  the  position  of  the  needle  be  as  shown. 
The  current  over  the  magnet  will  tend  to  drive  the  north 
pole  down  into  the  plane  of  the  coils,  the  south  pole  up  out 
of  the  plane,  as  shown.  Similarly,  the  current  below  the 


FIG.  177. 


332 


THEORY   OF  PHYSICS 


[CH.  IV 


magnet  will  have  identically  the  same  effect.  This  deflec- 
tion will  depend  upon  the  number  of  turns  of  wire,  and 
will  obviously  be  greater  for  an  increased  number.  It  will 
also  be  greater  for  a  greater  current.  Such  an  instrument 
as  this  is  called  a  "  galvanoscope,"  because  it  serves  to  de- 
tect even  a  minute  current. 

There  is  one  defect  in  the  instrument,  which  can  be  easily 
remedied,  however.  Owing  to  the  magnetic  action  of  the 
earth,  and  also  of  other  currents  and  magnets,  the  needle 
in  this  instrument  would  be  under  the  influence  of  other 

things  than  the  current  pass- 
ing around  its  own  coils.     To 
prevent  this,  a  rigid  combina- 
tion is  made  of  two  magnetic 
needles  as  nearly  identical  as 
N      possible,  but   placed   parallel 
1      with  their  poles  in  opposite 
directions.     It  is  evident  that 
S      no  external   magnet  or   cur- 
rent  would   have   any   influ- 
ence on  this  compound  needle, 

because  the  south  and  north  poles  are  so  close  together. 
Such  a  combination  is  called  an  "  astatic  system, "  because, 
if  perfect,  it  would  have 
no  tendency  to  point  north 
and  south,  or  in  any  defi- 
nite direction.  This  as- 
tatic needle  is  so  placed 
in  relation  to  the  coil  of 
wire  which  carries  the 
current  that  the  lower 
needle  is  inside  the  coil 
(just  as  one  was  in  the  FIG  ]79 

simpler   instrument),   the 

upper  one  is  just  above  the  coil.  As  may  be  readily  seen, 
the  action  of  the  current  on  this  upper  needle  is  also  to 


FIG.  178. 


254]    PROPERTIES  OF  STEADY  ELECTRIC  CURRENTS    333 

help  the  deflection.  This  magnetic  action  of  a  current 
will  be  more  fully  discussed  in  Chapter  VII. 

254.  Ohm's  Law.  If  there  is  a  steady  current  in  a  con- 
ductor, there  is  proved  by  direct  experiment  to  be  a  simple 
relation  between  the  intensity  of  the  current  and  the 
electromotive  force.  This  law  is  known  as  Ohm's  Law  ; 
and  it  may  be  stated  thus  : 
if  a  current  of  intensity,  i, 
is  flowing  in  a  conductor  ; 
arid  if  between  two  points,  Flo 

A  and  B,  of  that  conductor 

there  is  no  seat  of  E.  M.  P.  (such  as  a  junction  of  two  con- 
ductors or  a  solid  dipping  into  a  liquid  conductor),  —  there 
is  this  connection  between  the  difference  of  potentials  at 
A  and  B,  Ey  and  the  intensity  of  the  current,  it  viz.  : 

E/i  =  R     .......     (3) 

where  R  is  a  constant  depending  only  on  the  nature  of 
the  conductor  between  A  and  B.  E  is  known  as  the 
"  resistance  "  of  this  portion  of  the  conductor.  This  law 
states  that,  however  the  current  varies,  the  difference  of 
potential  E  varies  in  the  same  ratio  as  the  intensity. 
There  is  probably  no  law  of  nature  which  has  been  so 
completely  verified  by  experiment  as  this.  It  is  commonly 
written 


It,  the  resistance  of  the  conductor  between  A  and  B,  de- 
pends, as  has  been  said,  only  upon  the  nature  of  the  con- 
ductor, —  its  material  and  its  size.  If  either  of  these  is 
changed,  R  changes.  It  may  be  proved  by  experiment 
that,  if  the  conductor  is  of  uniform  cross-section,  <r,  and 
of  length  I,  the  value  of  R  may  be  written 


334 


THEORY  OF  PHYSICS 


[CH.  IV 


where  p  is  a  quantity  depending  only  upon  the  material  of 
the  conductor.  If  the  temperature  changes,  p  changes, 
too ;  it  increases  for  most  solid  conductors,  if  the  tempera- 
ture is  raised ;  and  decreases  for  most  liquid  conductors. 

The  reciprocal  of  the  resistance  is  called  the  conduct- 
ance, C.     Its  value  is,  then, 


(5) 


p  is  sometimes  called  the  "  resistivity  "  or  "  specific  resist- 
ance," and  -  is  also  called  the  "  conductivity  "  or  "  specific 

conductance." 

TABLE    XV 
SPECIFIC  CONDUCTIVITY,  REFERRED  TO  MERCURY 


Aluminium  (soft)    . 

32.35 

Nickel  (soft)  .     .  . 

3.14 

Copper  (pure)      .     . 

59. 

Platinum    .     .     . 

14.4 

9.75 

Silver  (soft)    . 

62.6 

Mercury 

1  000 

Tin         . 

7. 

KESISTANCE,  IN  OHMS,  OF  WIRE  100  CM.  LONG,  1  MM.  CROSS- 
SECTION 


• 

Rate  of  Change  in  Re- 
sistance per  degree 
Centigrade. 

Aluminium      

0  02916 

0  00388 

0.0160 

0  00380 

German  Silver      .... 
Iron         .     .           .... 

0.2670 
00973 

0.0035 
0  00650 

IVIercurv            

09434 

0  000907 

Platinum                . 

00907 

Silver      

0  01506 

0  00377 

These  values  in ( terms  of  "Ohms"  are  not  exactly  in 
the  unit  defined  later  in  Article  280.     The  Ohm  here  used 


256]    PROPERTIES  OF   STEA 


is  a  standard  resistance  which  differs  extremely  slightly 
from  the  true  Ohm  as  denned  later. 

APPLICATIONS  OF  OHM'S  LAW. 

255.    a.  Conductors  in  Series.     If  several  conductors,  A  P , 
B  C,  CD,  are  joined  in  series,  Ohm's  law  may  be  applied 
to  each  one  separately.    Let 
RI,  RZ,  Rs,  be  the  resistances 
of  the  conductors,  and  V^ 
VB,  etc.,  the  potentials  at 
A,  B,  etc.     Then,  since  the 

intensity  of  the  current  is  the  same  throughout  (see  Art. 
251), 


I  Z  * 

Hence,  if  there  is  no  E.  M.  F.  at  the  junctions  A,  B,  C, 


Therefore,  the  entire  resistance  between  A  and  D  is 

R  =.  Rl  +  R2  +  R3  .........     (6) 

That  is,  when  several  conductors  are  joined  in  series,  their 
entire  resistance  is  the  sum  of  the  resistances  of  the  sepa- 
rate conductors.  (This  shows  why  in  a  uniform  conductor 
R  is  proportional  to  its  length.) 

256.   5.  Conductors  in  Parallel.     If  a  conductor  branches 


FIG.  182. 


so  that  several  conductors  carry  the  current  between  two 
points,  they  are  said  to  be  in  parallel.     Thus,  let  three 


336  THEORY  OF   PHYSICS  [CH.  IV 

conductors,  of  resistances  RI,  R2,  Rz,  be  joined  in  parallel 
between  A  and  B  ;  and  let  the  currents  in  them  be  ii,  i2,  iy 
The  total  current 


and  the  entire  resistance  between  A  and  B  is  R  in  the 
formula 

T^ 


R 
But  i.=  VA~ 


12  = 


Hence  <  =  (VA  -  VB}       -  +     -  + 


^  -Kj  -Kg  -"3 

Expressed  in  terms  of  conductances,  the  total  conductance 
C  =a1+  <72+<73. 

That  is,  when  several  conductors  are  joined  in  parallel, 
the  total  conductance  equals  the  sum  of  the  conductances 
of  each  conductor.  (This  shows  why  R  varies  inversely 
as  the  cross-section  of  a  conductor.) 

Comparing  the  values  of  ii,  iv  and  is,  it  is  seen  that 


(8) 


..... 
i  •  i  =  —  •  —  I 


That  is,  when  a  current  is  divided  among  several  branches, 
the  intensities  in  any  two  branches  are  in  the  inverse  ratio 
of  their  resistances. 


257]    PROPERTIES  OF  STEADY  ELECTRIC   CURRENTS    337 

257.  c.  "  Wheatstone's  Bridge."  This  is  a  name  given  to 
a  particular  arrangement  of  conductors,  which  consists,  in 
principle,  of  two  conductors,  A  B  D  and  A  CD,  in  parallel, 
having  two  points,  B  and  C,  joined  by  a  third  conductor,  and 
having  a  fourth  conductor  joining  A  and  D  directly.  If  a 
primary  cell  is  placed  in  this  last  conductor,  AD,  a  cur- 


rent will  flow  either  to  A  or  to  D,  and  will  be  divided 
there  into  two  branches.     Call  the  resistance  of  the  con- 
ductor AB,E^  of  B  D,  EZ]  of  A  C,  R3  ;  of  CD,  Rt  ;  and 
let  the  respective  current  intensities  be  iit  i%,  is,  i±. 
Then,  by  Ohm's  law, 


. 


r-r 


. 


e 


B, 


There  will  in  general  be  a  current  through  the  conductor 
which  leads  from  B  to  C\  but,  if  there  is  none,  i.  e,  if 
VB  =  Vc,  then  ii  =  izt  and  i8  —  i±.  And  in  this  particular 
case  it  follows  from  the  above  equations  that 


or 


A 


(9) 


338 


THEORY  OF  PHYSICS 


[CH.  IV 


It  is  perfectly  easy,  though,  to  secure  the  condition  that 
VB  —  VQ.  If  a  galvanoscope  is  placed  in  the  conductor 
leading  from  B  to  C,  a  current  may  be  instantly  detected ; 
and  if  the  point  0  (or  B)  is  shifted  along  the  conductor 
A  C  D  (or  A  B  D),  it  must  be  possible  to  find  a  point  such 
that  no  current  flows  through  the  galvanoscope.  Then,  if 
J?3  and  R±  are  the  resistances  between  this  point  C  and  A 
and  D,  respectively,  the  above  equation  is  true. 

Consequently,  if  the  numerical  values  of  three  resist- 
ances are  known,  that  of  the  fourth  may  be  determined. 
The  unit  of  resistance  will  be  denned  later  (see  Art.  279) ; 
but  Wheats  tone's  bridge  gives  at  once  a  method  of  com- 
paring resistances. 


o 


-£J         19 

KAAA! 


FIG.  184. 

A  simple  modification  of  the  bridge  is  to  make  the  con- 
ductor, A  CD,  a  long  uniform  wire.  In  which  case,  if  13 
and  /4  are  the  lengths  of  the  wire  from  A  to  C  and  from  C 
toZ>, 

JRS  •  R±  =  Is '  l±  j 


and  so,  when  no  current  flows  from  C  to  B, 
RI  :  HZ  =B  H  *  N     • 


(10) 


The  lengths  13  and  74  may  be  measured ;  and  thus  the  ratio 
of  any  two  resistances  may  be  easily  determined.  Fuller 
details  of  the  method  are  given  in  all  laboratory  manuals ; 


258]    PROPERTIES  OF  STEADY  ELECTRIC   CURRENTS     339 

and  so  are  also  other  methods  for  the  comparison  of 
resistances. 

258.  d.  Heating  Effect.  As  proved  in  Article  252,  the  heat- 
ing effect  in  any  portion  of  a  conductor  through  which  a 
current  of  intensity  i  flows  for  t  seconds  is  E  i  t,  if  E  is  the 
difference  of  potential  at  the  two  ends  of  the  conductor. 
By  Ohm's  law,  if  the  current  is  steady,  E  =  i  R.  Hence, 
substituting  for  E  in  equation  (1)  its  value, 

H=PRt (11) 

This  shows  that,  if  a  current  is  flowing  through  any  series 
of  conductors,  the  heat  effect  is  greatest  where  the  resist- 
ance is  greatest.  This  explains  the  great  rise  in  tempera- 
ture wherever  there  is  a  bad  contact  in  the  circuit. 

This  shows,  too,  that  the  heating  effect  through  a  con- 
ductor is  independent  of  the  direction  of  the  current ;  for, 
if  the  current  is  reversed,  the  intensity  becomes  —  i,  and 
H  =  i2  R  t,  as  before. 

This  same  formula  may  be  expressed  also  in  terms  of  E 
and  R ;  for,  substituting  i  =  E /  R,  it  becomes 

H=  E*t/R (11  a) 


CHAPTEE  V 


GENERAL  PROPERTIES  OF  MAGNETS  AND  MAGNETIC 
FIELDS 

IT  is  well  known  to  every  one  that  an  ordinary  magnet 
has  the  power  of  attracting  pieces  of  iron  or  steel;  and 
that,  if  a  light  magnetic  needle  is  pivoted  so  as  to  be  free 

to  turn,  it  will  tend  to  turn 
into  a  direction  nearly  north  *. 
and  south.  It  is  also  probably 
known  to  all  that  there  is  an 
intimate  connection  between  an 
electric  current  and  a  magnet, 
as  is  illustrated  in  the  last 
chapter  (Art.  253).  Before  dis- 
cussing this  last  property  of 
a  magnet,  it  will  be  necessary 
to  consider  in  detail  the  first  two. 

259.  Definitions.  A  magnet  may  be  defined  as  a  body 
which,  without  being  electrified,  has  the  power  of  attracting 
pieces  of  iron  in  its  natural  state.  This  means  that  work 
is  required  to  move  a  piece  of  iron  away  from  a  magnet, 
and  that  consequently  the  potential  energy  will  be  de- 
creased if  the  iron  and  the  magnet  approach  each  other. 
So  they  tend  to  approach,  and  will  do  so  unless  hindered 
by  some  external  force.  This  energy  which  is  associated 
with  magnets  is  in  the  ether ;  and  the  presence  of  ordi- 
nary matter  influences  the  phenomenon,  —  for  example, 
the  forces  in  air  are  entirely  different  from  those  in  pure 
oxygen.  But  when  magnets  are  near  each  other,  there  is 


FIG.  185. 


260] 


MAGNETS  AND  MAGNETIC  FIELDS 


341 


no  such  statical  mechanical  strain  of  the  surrounding 
medium  as  there  is  in  the  case  of  charged  bodies.  In  fact, 
as  will  be  more  clearly  seen  further  on,  the  phenomena  of 
magnetism  are  evidently  conditioned  by  motion  in  the 
ether,  and  in  the  smallest  portions  of  the  surrounding 
medium. 

Other  substances  than  iron  are  attracted  by  magnets 
when  the  surrounding  medium  is  air ;  among  them  are 
nickel,  cobalt,  manganese,  and  many  compounds  of  iron. 
Such  bodies  are  called  "  magnetic  substances." 

Other  bodies  are  repelled  by  magnets,  when  the  sur- 
rounding medium  is  air;  such  are  bismuth,  antimony, 
and  zinc.  All  such  bodies  are  called  "diamagnetic  sub- 
stances ; "  but  the  forces  of  repulsion  are  extremely  minute 
compared  with  the  forces  of  attraction  of  a  magnet  for  iron. 

Attraction  or  repulsion  will,  of  course,  take  place,  so  that 
the  potential  energy  may  decrease.  (See  Art.  221  for  the 
similar  case  in  electricity.)  The  motion  in  air  may,  there- 
fore, be  different  from  that  in  other  media ;  the  question  is 
one  simply  of  relative  amounts  of  po- 
tential energy  in  equal  volumes  of  dif- 
ferent media. 

260.  Magnets.  There  is  one  natural 
substance  which  is  a  magnet,  viz.  the 
so-called  magnetic  oxide  of  iron  (Fe304), 
which  occurs  as  an  ore  in  many  parts 
of  the  world.  But  it  is  produced  in 
lumps,  and  so  is  not  generally  useful. 
Any  magnetic  substance,  such  as  steel, 
iron,  etc.,  may,  however,  be  made  a  mag- 
net ;  and  convenient  shapes,  such  as 
cylinders,  bars,  and  needles,  may  be 
chosen.  The  simplest  way  of  making 
a  bar  of  iron  or  steel  a  magnet  is  to 
place  it  in  a  helix  or  spiral  coil  of  wire  through  which  an 
electric  current  is  passing.  (The  current  passes  around 


FIG.  186. 


342  THEORY   OF  PHYSICS  [CH.  V 

the  bar  of  iron,  not  through  it.)  If  this  is  done,  the  bar  is 
found  to  be  a  magnet,  and  remains  so  when  removed  from 
the  current.  If  the  material  is  soft  iron,  the  magnetism 
will  disappear,  however,  at  the  least  disturbance,  such  as 
tapping  it ;  whereas,  if  the  bar  is  steel,  it  will  remain  a 
magnet  for  a  long  time,  and  is  sometimes  called  a  "  per- 
manent "  magnet.  (The  fuller  explanation  of  this  method 
of  making  a  magnet  will  be  given  later,  in  Chapter  VII.) 
If  such  a  permanent  magnet  is  once  secured,  it  can  be  used 
to  make  other  magnets  ;  for,  if  this  magnet  is  drawn 
lengthwise  over  a  bar  of  iron  or  steel,  the  latter  itself 
becomes  a  magnet.  If  a  bar  of  iron  or  steel  is  even  placed 
near  a  permanent  magnet,  it  becomes  itself-  a  magnet, 
although  a  weak  one. 

261.  Polarity.  If  a  bar  or  needle  of  steel  is  magnetized 
by  being  placed  along  the  axis  of  a  helix  of  wire  carrying  a 
current,  it  is  said  to  be  magnetized  "  lengthwise  ; "  and  if, 
after  being  removed  from  the  helix,  it  is  pivoted  or  sus- 
pended 'free  to  turn,  it  will  place  itself  in  a  direction  nearly 
north  and  south,  in  what  is  called  the  "  magnetic  meridian." 
This  shows  that  the  two  ends  of  the  magnet  have  opposite 
properties  :  one  is  attracted  towards  the  north,  the  other  is 
repelled  from  the  north.  The  bar  or  needle  is  said  to  be 
"  polarized ; "  and  the  end  towards  the  north  is  called  the 
"  north  pole  "  of  the  magnet ;  that  towards  the  south,  the 
"south  pole." 

If  the  action  of  one  bar-magnet  on  another  is  ex- 
amined, it  is  seen  that  two  like  poles  repel  each  other, 
while  two  unlike  poles  attract  each  other.  (It  must  not  be 
thought  that  all  magnets  have  poles ;  it  is  true  only  of  bar- 
magnets  and  needles,  in  general.)  The  exact  position  of 
the  "  pole,"  considered  as  a  point,  is  not  definite.  If  the 
forces  in  the  neighborhood  of  a  bar-magnet  are  examined 
by  means  of  a  small  magnetic  needle,  it  may  be  proved 
that  there  are  magnetic  forces  at  all  points  of  the  bar- 
magnet;  so  that,  if  the  bar-magnet  is  placed  at  right 


263]  MAGNETS   AND   MAGNETIC  FIELDS  343 

angles  to  the  magnetic  meridian,  there  is  a  series  of  parallel 
forces  attracting  one  half  the  magnet  towards  the  north, 
and  another  series  of  parallel  forces  repelling  the  other  half. 
The  resultants  of  each  of  these  two  sets  of  parallel  forces 
are  two  parallel  forces ;  and  they  meet  the  magnet  in  two 
points  which  may  be  called  its  "poles."  The  distance 
between  these  two  poles  is  called  the  "length"  of  the 
magnet. 

262.  Equality  of  Poles.     If  a  bar-magnet  is  placed  on  a 
cork  which  floats  in  water,  so  that  it  is  free  to  move  in  any 
way,  it  is  observed  that  the  magnet  does  not  move  as  a 
whole  in  any  direction.    Its  centre  of  inertia  remains  fixed; 
and  it  turns  around  an  axis  through  this  until  it  comes  to 
rest  in  the  magnetic  meridian.     This  proves  that  the  forces 
on  the  two  poles  are  exactly  equal  and  opposite;  or,  in 
other  words,  the  poles  are  of  equal  "  strengths." 

If  any  other  magnet,  even  a  lump  of  magnetic  ore,  is 
placed  on  the  cork,  it  also  will  have  no  motion  of  transla- 
tion, showing  that  all  the  forces  due  to  the  magnetic  field 
of  the  earth  reduce  to  a  couple. 

263.  Molecular  Nature   of    Magnetism.      There   is  every 
reason  for  believing  that  magnetism  is  a  molecular  property 
of  a  body ;  that  every  molecule  of  a  magnet  is  itself  a 
magnet.     For,  anything  which  influences  the  molecules  of 
a  magnet  affects  the  magnetism  as  well.     Thus,  increase  in 
temperature  decreases  the  magnetism ;  and  at  "  red-heat " 
a  steel  magnet  loses  practically  all  its  magnetism.     Ham- 
mering a  magnet  will  always  alter  its  magnetism.    Further, 
if  a  magnet  is  broken  in  two,  each  portion  is  found  to  be  a 
magnet,  no  matter  how  minute  the  fragments  are. 

Since  any  magnetic  substance  (iron,  steel,  nickel,  etc.) 
may  be  made  a  magnet,  the  entire  phenomena  of  magneti- 
zation may  be  explained  if  each  molecule  of  a  magnetic 
substance  is  a  magnet  with  equal  poles.  In  a  bar  of  such 
a  substance  the  molecules  may  be  considered  as  lying  at 
random,  and  therefore  neutralizing  each  other's  external 


344  THEORY  OF  PHYSICS  [CH.  V 

action  j  and  the  process  of  making  such  a  bar  a  magnet 
consists  simply  in  turning  the  molecular  magnets  into 
more  or  less  the  same  direction.  Tor,  if  this  is  done,  the 
north  poles  being  in  one  direction  and  the  south  poles  in 
the  other,  there  will  be  a  resultant  action  near  the  two  ends 
of  the  bar.  (The  molecules  in  the  figure  are  drawn  elon- 
gated, simply  to  liken  them  to  ordinary  bar-magnets,  not  to 


^^   — ^c^    ^    ^   S  S   /    \ 

FIG.  187. 

imply  that  a  molecule  is  like  that.)  This  explains  also 
why  the  two  poles  of  a  bar-magnet  are  equal,  because  each 
molecular  magnet  has  equal  poles.  Similarly,  even  if  the 
magnet  is  not  in  the  shape  of  a  bar  or  needle,  but  has 
minute  poles  all  over  it,  it  is  seen  that  the  north  poles 
must  equal  the  south  poles  in  strength. 

It  is  evident,  too,  why  a  bar  of  iron  becomes  a  magnet 
when  a  permanent  magnet  is  rubbed  over  it  lengthwise  or 

even  placed  near  it,  because  that 

|N  Sj   pole   of   the   permanent   magnet 

which  passes  over  the  iron,  or 
which  is  nearest  it,  attracts  all 
the  opposite  poles  of  the  molecu- 
lar magnets  of  the  iron,  and  re- 
pels all  the  like  poles,  thus  turn- 
188.  ing  the  molecules  into  a  more  or 

less   parallel  direction.     If   they 

are  all  forced  into  a  strictly  parallel  direction,  the  magneti- 
zation must  be  the  greatest  possible.  It  is  observed,  in 


264]  MAGNETS  AND  MAGNETIC  FIELDS  345 

fact,  that  a  bar  of  iron  or  any  magnetic  substance  cannot 
be  magnetized  beyond  a  definite  limit,  in  which  state  it  is 
said  to  be  "  saturated." 

Another  evidence  in  proof  of  the  fact  that  each  mole- 
cule of  a  magnetic  substance  is  a  magnet  is  the  fact  that  it 
is  possible  to  make  a  model  of  such  a  substance  which 
exhibits  all  its  properties.  If  an  immense  number  of 
small  pivoted  magnets  are  placed  near  each  other  on  a 
table,  they  will  behave  exactly  as  do  the  molecules  of  a 
magnetic  substance.  At  first  they  will  stand  perfectly  at 
random ;  then,  if  a  current  is  passed  around  them  by  plac- 
ing the  table  in  a  helix  of  wire,  or  if  a  strong  magnet  is 
brought  near,  the  small  magnets  will  place  themselves 
more  or  less  parallel ;  and,  if  the  current  is  stopped  or  the 
magnet  removed,  the  small  magnets  will  still  remain  nearly 
parallel,  unless  disturbed  by  jarring. 

A  further  evidence  is  afforded  by  the  fact  that  when  a 
bar  of  a  magnetic  substance  is  magnetized  suddenly,  a  dis- 
tinct click  is  heard,  known  as  the  "  Page  effect ;  "  and  the 
length  and  volume  of  the  bar  also  change  (except  in  quite 
special  cases). 

Since,  then,  magnetism  is  a  molecular  property,  and  each 
molecule  of  a  magnetic  substance  has  both  a  north  pole 
and  a  south  one,  it  is  evident  that  a  north  pole  cannot  be 
separated  from  its  equal  south  pole ;  the  same  magnet 
must  always  have  both  north  and  south  poles.  (This  is 
unlike  electrical  charges,  which  can  be  separated  so  as  to 
be  on  different  bodies.) 

261  "  "Unit  Poles."  Two  equal  poles  may,  of  course,  be 
defined  as  two  poles  which  have  identically  the  same 
effects,  e.  g.  exert  the  same  forces  on  a  suspended  magnet. 
And,  if  two  equal  poles  when  placed  one  centimetre  apart 
in  air,  exert  a  force  on  each  other  of  one  dyne,  they  are 
each  said  to  be  "  unit  poles  ;  "  or  each  pole  is  said  to  have 
a  "  strength  "  one. 

Law    of   Force.     The   law   of   action   between   magnets 


346  THEORY  OF  PHYSICS  [CH-  V 

may  be  stated  in  this  way:  if  a  pole  of  strength  m  is 
placed  at  a  distance  of  r  cm.  from  another  pole  of  strength 
TO'  in  any  medium,  the  force  in  dynes  is  given  by  the 
formula 


F- 


where  JJL  is  a  quantity  which  depends  upon  the  medium. 
It  is  not,  in  general,  constant  for  any  one  medium,  but  varies 
with  the  amount  of  the  magnetic  force.  It  has  received 
the  name  "permeability,"  for  reasons  which  will  be  given 
later.  This  law  of  force  is  amply  verified  by  experiment, 
because  all  deductions  from  it  are  in  accordance  with 
observed  phenomena. 

For  air  IJL  is  constant  ;  and  it  equals  1,  on  the  above  defi- 
nition of  a  unit  pole.  For,  if  m  =  mf  —  1,  and  r  —  1  in 
air,  F  must  equal  1  ;  i.  e.  p  =  1  for  air.  If  a  north  pole 
has  a  strength  m,  an  equal  south  pole  must  have  a  strength 
—  m  ;  because  the  force  on  a  north  pole  is  exactly  opposite 
to  that  on  an  equal  south  pole. 

For  all  magnetic  substances  p  is  greater  than  1,  on  the 
magnetic  system  of  units  above  defined  ;  and  for  all  dia- 
magnetic  substances  /JL  is  less  than  1.  It  should  be  noticed, 
though,  that,  since  all  diamagnetic  forces  are  so  feeble,  p 
may  in  practice  be  regarded  as  being  equal  to  1  except  in 
magnetic  media.  These  facts  may  be  proved  from  analogy 
with  similar  facts  in  electricity,  as  follows. 

265.  Attraction  and  Repulsion.  K,  the  dielectric  constant 
in  electricity,  plays  the  same  part  in  electrical  phenomena 
that  p,  the  magnetic  permeability,  does  in  magnetic.  But 
it  was  proved  in  Chapter  I.  Article  221,  that,  if  any  piece 
of  dielectric,  e.  g.  a  bit  of  glass  or  sulphur,  is  attracted  by  a 
charged  body  in  air  (where  K  =  1),  this  is  evidence  that 
K  for  that  dielectric  is  greater  than  for  air  ;  whereas,  if  a 
piece  of  dielectric  is  repelled,  K  for  it  is  less  than  for  air. 
But  in  magnetic  phenomena  magnetic  bodies  are  attracted 


266]  MAGNETS   AND   MAGNETIC  FIELDS  347 

in  air  by  a  magnet,  and  diamagnetic  bodies  are  repelled. 
Consequently,  p  for  the  former  is  greater  than  for  air,  and 
for  the  latter  is  less. 

266.  Magnetic  Lines  of  Force.  A  magnetic  field  of  force 
may  be  defined  as  the  region  in  which  magnetic  forces  are 
manifest ;  and  magnetic  lines  of  force  may  be  drawn  just 
as  electric  lines  of  force  were,  viz.  a  magnetic  line  of  force 
gives  at  each  point  the  direction  in  which  a  north  pole 
would  move  if  placed  at  that  point.  A  south  pole  would, 
of  course,  tend  to  move  in  the  opposite  direction ;  and,  con- 
sequently, if  a  very  small  magnet  is  placed  at  any  point  in 
a  magnetic  field,  it  will  place  itself  tangent  to  the  line  of 
force  at  that  point.  (In  this  way  the  lines  of  force  of  any 
magnetic  field  may  be  mapped.)  If  the  lines  of  force  are 
all  parallel,  the  field  is  said  to  be  " uniform." 

Lines  of  magnetic  force  always  pass  from  north  poles  to 
south  poles ;  and,  since  a  magnet  is  made  up  of  small  mag- 


FIG.  189. 


nets,  lines  of  force  pass  through  magnets,  and  do  not  end  on 
their  surfaces.  A  drawing  is  given  of  the  lines  of  force 
around  a  bar  magnet.  Magnetic  "potential"  at  a  point 
in  the  field  may  be  defined  as  the  work  required  to 
bring  up  a  unit  north  pole  from  infinity  to  that  point.  (The 
earth  cannot  in  this  case  be  taken  as  the  standard,  because 
its  magnetic  potential  is  not  the  same  at  all  points.)  So 
equipotential  surfaces  for  magnetism  may  be  drawn ;  and 
lines  of  magnetic  force  are  always  perpendicular  to  them, 
and  pass  from  high  to  low  potential. 


348  THEORY  OF  PHYSICS  [CH.  V 

The  "  intensity "  of  the  magnetic  force  at  any  point  of 
the  field  is  the  name  given  to  the  force  which  would  act  on 
a  unit  north  pole  if  placed  at  that  point ;  and  wherever  the 
lines  of  force  are  thickest  the  intensity  is  greatest,  as  is 
shown  by  the  lines  around  a  magnet. 

267.  Magnetic  Induction.  When  any  magnetic  substance 
is  placed  near  a  magnet,  i.  e.  is  placed  in  a  magnetic  field, 
two  phenomena  are  observed,  as  already  noted :  it  becomes 
a  magnet,  and  it  is  attracted  by  the  permanent  magnet. 
These  phenomena  have  been  already  explained  as  conse- 
quences of  the  fact  that  the  molecules  of  a  magnetic  sub- 
stance are  themselves  magnets ;  for,  if  the  north  pole  of  the 
permanent  magnet  is  nearer  the  magnetic  substance  than 
the  south  pole,  the  south  poles  of  the  molecular  magnets 
will  be  turned  towards  it ;  and  so  there  will  be  attraction. 
This  phenomenon  is  known  as  magnetic  "  induction."  The 


FIG.  190. 


simplest  case  is  when  a  bar  of  some  magnetic  substance  is 
placed  lengthwise  in  a  uniform  field  of  force  in  air.  The 
field  of  force  is  found  to  be  changed  as  shown.  The  lines 
of  force  crowd  down  into  the  magnetic  substance,  out  of 
the  air.  (Compare  Art.  229  and  Fig.  158  for  the  similar 
case  in  electricity.)  This  phenomenon  may  be  described 
by  saying  that  the  magnetic  substance  is  more  permeable 
for  lines  of  magnetic  force  than  air ;  and,  since  p  is  greater 
for  it  than  for  air,  p  is  called  the  permeability.  This 
change  in  the  lines  of  force  is  exactly  the  same  as  would 
take  place  if  there  were  superimposed  upon  the  original 


267]  MAGNETS  AND  MAGNETIC  FIELDS  .      349 

uniform  field  that  due  to  a  permanent  magnet  of  suitable 
strength  and  of  the  same  size  and  shape  as  the  bar  of  mag- 
netic material,  with  its  south  pole  placed  at  the  end  into 
which  enter  the  lines  of  force  due  to  the  uniform  field. 
For,  at  any  point,  P,  there  would  be  two  forces,  as  shown, — 


N|  |S 


FIG.  191. 

one  due  to  the  uniform  field,  the  other  to  the  magnet ;  and 
their  resultant  has  the  same  direction  as  the  line  of  force 
actually  observed  when  the  bar  of  magnetic  material  is 
placed  in  the  uniform  field.  If  the  field  of  force  is  uni- 
form inside  the  bar,  as  it  will  be  near  its  middle  point  if 
the  bar  is  very  long,  a  small  thin  cavity  may  be  imagined 
cut  across  the  bar,  or  the  bar  may  be  imagined  cut  in  two 
and  the  parts  separated  slightly ;  and  the  intensity  of  the 
magnetic  force  in  that  cavity  is  called  the  "  magnetic  in- 
duction." Its  ratio  to  the  intensity  of  the  magnetic  force 
in  the  original  uniform  field  in  air  may  be  proved  to  be 
the  value  of  p  for  the  magnetic  substance. 


FIG.  192. 


Similarly,  if  a  diamagnetic  substance  is  placed  in  a  uni- 
form field,  the  modification  in  the  lines  of   force   is  as 


350  THEORY  OF  PHYSICS  [CH.  V 

shown ;  the  lines  crowd  out  into  the  air ;  the  diamagnetic 
substance  is  less  permeable.  This  change  is  exactly  the 
same  as  would  take  place  if  there  was  superimposed  upon 


FIG.  193. 

the  original  uniform  field  that  due  to  a  magnet  of  suitable 
strength  and  of  the  same  size  and  shape  as  the  diamag- 
netic substance,  but  with  its  north  pole  at  the  end  into 
which  enter  the  lines  of  force  of  the  uniform  field.  This 
is  shown  in  the  drawing. 

Of  course  the  attractions  and  repulsions  take  place  in 
accordance  with  the  general  law  that  the  potential  energy 
tends  to  decrease.  If  a  magnetic  substance  is  placed  any- 
where in  the  field,  there  will  be  less  energy  in  it  than  in 
an  equal  volume  of  air  at  that  same  point  in  the  field  ; 
and,  the  more  intense  the  field,  the  greater  the  difference 
in  the  energy  between  the  magnetic  substance  and  the  air. 
Consequently,  a  magnetic  substance  will  move  into  that 
portion  of  the  field  where  the  intensity  is  greatest,  while  a 
diamagnetic  substance  will  move  into  that  portion  where 
the  field  is  weakest.  In  a  uniform  field,  neither  would 
move. 

Since  a  magnetic  substance  is  more  permeable  to  lines 
of  force  than  air,  if  a  hollow  sphere  of  iron  is  placed 
in  a  field  of  force,  there  will  be  comparatively  no  lines 
of  force  inside,  because  nearly  all  will  pass  through 
the  iron  shell.  Similarly,  if  an  iron  ring  is  placed  in 
a  uniform  field,  the  lines  of  force  will  be  as  shown,  the 
field  being  most  intense  at  the  top  and  bottom  of  the 
ring. 


268] 


MAGNETS   AND  MAGNETIC   FIELDS 


351 


FIG.  194. 

268.  Magnetic  Moment.  If  any  magnet  is  placed  at 
random  in  a  magnetic  field,  and  is  suspended  so  as  to  be 
free  to  rotate  around  the  axis  of  suspension,  it  will  in  gen- 
eral turn  and  tend  to  place  itself  in  some  particular  posi- 
tion, i.  e.  there  is  a  mechanical  moment  acting  on  it  due  to 
the  field  of  force.  The  simplest  case  is  when  the  field  is 
uniform,  and  the  intensity  at  each  point  is  1.  Such  a  field 
is  called  a  "  unit  field."  The  greatest  value  that  the  me- 
chanical moment  can  have,  which  acts  on  the  magnet 
when  placed  in  any  position  in  this  unit  field,  is  called  the 
"  magnetic  moment  "  of  the  magnet.  The  simplest  magnet 
is  a  bar-magnet  with  its  two  poles  of  strength,  m,  at  a  dis- 
tance, I,  apart.  The  maximum  mo- 
ment which  it  can  experience  in  a  — 
unit  field  is  obviously  that  which  it 
has  when  it  is  at  right  angles  to  the  _ 
field.  There  will  then  be  a  force  of 
m  dynes,  parallel  to  the  field,  acting 
on  its  north  pole,  because  by  defini- 
tion of  intensity  it  is  the  force  act- 
ing on  a  unit  pole ;  and,  if  the  intensity  is  1,  the  force 
on  a  pole  of  strength  m  is  m  dynes.  Similarly,  there  will 
be  a  force  —  m  dynes  acting  on  its  south  pole  ;  i.  e.  a  force 
exactly  opposite  to  the  one  on  the  north  pole.  These  two 
forces  form  a  couple ;  and,  since  the  distance  between  the 


FIG.  195. 


352 


THEORY   OF  PHYSICS 


[CH.  V 


poles  is  I,  the  moment  of  the  couple  is  m  I.     If  the  mag- 
netic moment  is  written  M,  then  for  a  bar-magnet 

M=ml (2) 

If  a  bar-magnet  is  placed  at  an  angle  0  with  the  direc- 
tion of  the  unit  field  of  force,  the  moment  acting  on  it 
is  M  sin  0,  because  the  distance  be- 
tween the  two  parallel  forces  is  now 
/  sin  0.  If  the  uniform  field  of  force 
has  an  intensity  E,  the  force  acting  on 
the  pole  m  is  m  E  dynes ;  and  so  when 
the  bar-magnet  is  placed  at  an  angle  6 
with  the  direction  of  such  a  field,  the 
mechanical  moment  is  E  M  sin  6. 

269.  Measurement  of  R  M.  When  a 
magnet  is  not  parallel  to  the  field  of 
force,  there  is  a  couple  acting  on  it 
tending  to  make  it  turn  into  that  di- 
rection. So,  if  a  magnet  is  suspended 
free  to  turn  about  an  axis  perpendicu- 
lar to  the  field,  and  is  displaced  slightly  out  of  the  direction 
of  the  field  of  force,  it  will  be  turned  back  into  that  direc- 
tion, and  will  swing  through  it,  then  return,  etc.,  making 
oscillations  until  it  is  brought  to  rest  by  friction.  The 
mechanical  moment  at  any  instant  is  E  M  sin  6  ;  or,  if  the 
angle  is  very  small,  E  M  0.  This  moment  is  proportional 
to  0,  the  angular  displacement ;  and  it  tends  to  bring  the 
magnet  back  into  the  direction  of  the  field  of  force  ;  con- 
sequently, the  motion  is  simple  harmonic  vibration  (see 
Art.  25) ;  and,  if  A  is  the  moment  of  inertia  of  the  mag- 
net around  the  pivot  as  an  axis,  the  period  of  one  complete 
vibration  is 

T=27r^£M'  '  '  '  '/  (3) 
for  the  angular  acceleration  is  — - — ,  the  moment  of  the 
forces  divided  by  the  moment  of  inertia. 


FIG.  196. 


270]  MAGNETS  AND  MAGNETIC  FIELDS  353 

A          9      A 

Consequently     EM—  (3  a) 

T,  the  period,  can  always  be  observed ;  and  A  can  in  many 
cases  be  easily  calculated  from  a  knowledge  of  the  mass 
and  dimensions  of  the  magnet ;  so  E  M  may  be  determined. 
If  the  same  magnet  is  allowed  to  vibrate  in  succession 
in  two  fields  of  force,  whose  intensities  are  R\  and  Ez ;  and 
if  the  periods  of  vibration  are  T\  and  Tz, 


7*2 

Hence  RiiR^-TfiTf (4) 

So  by  means  of  a  vibrating  magnet  it  is  possible  to  com- 
pare the  intensities  of  two  fields  of  force. 

270.  Measurement  of  M  /  E.  If  this  same  bar-magnet, 
whose  period  has  been  determined  when  vibrating  freely 
in  a  uniform  field  of  intensity,  R,  is  placed  at  rest  at  right 
angles  to  the  field,  it  itself  will  produce  a  field  of  force 
which  at  some  distance  away  is  nearly  uniform,  and  at 
right  angles  to  the  existing  field,  R,  So,  if  a  small  mag- 
netic needle  is  suspended  at  a  point  some  distance  away  in 
the  direction  of  the  length  of  the  bar-magnet  in  such  a 
manner  as  to  be  free  to  turn  around  an  axis,  perpendicular 
to  the  field  of  force  and  to  the  bar-magnet,  it  will  be  under 
the  influence  of  two  fields  of  force  at  right  angles  to  each 
other.  It  will  therefore  place  itself  at  such  an  angle  that 
the  moments  due  to  the  two  fields  are  equal  but  oppo- 
site, so  that  they  neutralize  each  other.  Call,  for  the  time 
being,  the  intensity  of  the  field  due  to  the  bar-magnet,  /. 
The  moment  due  to  the  field  whose  intensity  is  R,  when 
the  magnet  makes  with  its  direction  the  angle  a,  is 
R  M'  sin  a,  if  Mr  is  the  magnetic  moment  of  the  magnetic 

12 


354 


THEORY  OF  PHYSICS 


[CH.  V 
R 


FIG.  197. 

needle.  But  the  magnet  makes  the  angle  90°  —  a  with 
the  field  of  intensity  /;  hence  the  moment  due  to  it  is 
f  M'  cos  a.  These  moments  are  in  opposite  directions, 
and  must  be  equal,  if  the  needle  is  at  rest. 

Hence               R  Mr  sin  a  =  f  Mr  cos  a, 
or  tana=//.R (5) 

The  value  of  /  may,  however,  be  easily  expressed  in 
terms  of  M,  the  magnetic  moment  of  the  large  bar-magnet 
and  r  the  distance  from  the  centre  of  this  magnet  to  the 
centre  of  the  magnetic  needle,  /  is  the  intensity  at  the 
point  0,  the  centre  of  the  needle ;  i.  e.  it  is  the  force 
which  would  act  on  a  unit  north  pole  if  placed  there.  The 
large  bar-magnet,  as  placed  in  the  diagram,  has  a  pole  of 

strength  +  m  at  a  distance  r  —  ^  away  from  0,  and  another 

I 

of  strength  —  m  at  a  distance  r  +  ~  •     Hence  the  force  on 

a  unit  north  pole  at  0  is 

2mlr 


/  = 


m 


m 


But,  if  r  is  very  great  in  comparison  with  /, 

2  ml r      2  M 


(6) 


271]  MAGNETS  AND  MAGNETIC  FIELDS  355 

Consequently,  substituting  in  (5), 

2M 


M       r3  tan  a 

=        - 


Various  precautions  and  modifications  for  this  experiment 
are   explained  in   laboratory  manuals,   but   it  is   evident 

that   both  r  and  a  can  be  measured;   and  so  --  may  be 

H 

determined. 

271,   Measurement  of  E  or  M.     By  a  combination  of  the 
two  formulae  (3  a)  and  (7),  it  is  seen  that 


and  so  both  R  and  M  may  be  measured. 

If  R  is  known  for  any  one  field,  it  has  been  explained 
how  its  value  for  any  other  field  can  be  determined  by 
means  of  a  vibrating  magnet  whose  period  can  be  measured. 


UNIVERSITY  OF  CALIFORNIA 

DEPARTMENT  OF  PHYSICS 


CHAPTEE  VI 

MAGNETISM  OF  THE  EARTH 

THAT  there  is  a  magnetic  field  of  force  around  the 
earth  is  proved  by  the  fact  that  a  suspended  magnet 
always  tends  to  take  a  definite  position  with  reference  to 
it.  If  a  magnetic  needle  is  suspended  so  as  to  be  perfectly 
free  to  turn  around  a  vertical  and  also  a  horizontal  axis,  it 
will  not  come  to  rest  in  a  horizontal  plane,  but  will  be 
inclined  to  it  by  a  certain  angle.  The  direction  taken  by 
the  needle  is,  of  course,  the  direction  of  the  line  of  mag- 
netic force  at  that  point  due  to  the  earth.  To  describe 
this  direction,  it  is  necessary  to  know  how  far  east  or  west 
of  the  true  geographical  north  and  south  line  the  magnet 
points ;  and  also  what  angle  the  magnet  makes  with  the 
horizontal  plane. 

Declination.  The  plane  which  passes  through  the  centre 
of  the  earth  and  the  magnetic  needle  when  it  has  come  to 
rest  is  called  the  "  magnetic  meridian ; "  and  the  angle  be- 
tween this  plane  and  the  geographical  meridian  at  the 
point  where  the  magnet  is  suspended  is  called  the  mag- 
netic "declination  "  at  that  point. 

It  is  determined  by  suspending  a  magnetic  needle  so 
that  it  is  free  to  turn  around  a  vertical  axis,  and  by  then 
measuring  the  angle  between  the  direction  of  the  needle 
and  the  geographical  north  and  south  line.  A  horizontal 
line  at  right  angles  to  the  magnetic  meridian  is  said  to 
have  the  direction  "  magnetic  east  and  west." 


272]  MAGNETISM   OF  THE   EARTH  357 

Inclination  or  Dip.  The  angle  made  by  the  horizontal 
plane  and  the  needle,  which  is  perfectly  free  to  turn,  is 
called  the  magnetic  "inclination,"  or  "dip." 

It  is  determined  by  suspending  a  magnet  on  a  horizontal 
axis  which  is  placed  perpendicular  to  the  magnetic  merid- 
ian, and  then  measuring  the  angle  between  the  magnet  and 
the  horizon. 

Intensity.  The  earth's  magnetic  H  _  ^ 
field  has,  at  each  point  on  the  sur- 
face, a  certain  intensity  ;  and  the 
direction  of  the  force  is  given  by 
the  declination  and  dip.  The  force 
at  any  point  may  be  resolved  into 
two  components,  —  one  vertical,  the 
other  horizontal.  If  6  is  the  dip,  FlG 

R  the  intensity  of  the  earth's  force, 

H  and   V  the  horizontal  and  vertical  components  of  K, 
then 


So  that,  if  6  and  H  are  known,  R  can  be  calculated. 
Similarly, 

tan  6=  VI  H    .......     (la) 

and,  if  this  ratio  V  /  H  can  be  measured,  6,  the  dip,  is  at 
once  determined. 

It  is  perfectly  possible  to  measure  II,  the  horizontal 
component  of  the  intensity  of  the  earth's  field,  by  the 
method  described  in  the  last  chapter  (Art.  271).  It  is 
simply  necessary  to  perform  two  experiments  :  measure 
the  period  of  a  bar-magnet  vibrating  about  a  vertical  axis  ; 
measure  the  horizontal  deflection  of  a  small  magnetic 
needle  which  is  produced  by  the  first  magnet  when  it  is 
placed  magnetic  east  or  west  of  the  needle. 

Earth's  Elements,  Variations.  The  declination,  dip,  and 
intensity  of  the  magnetic  field  at  any  point  on  the  earth's 


358  THEORY   OF  PHYSICS  [CH.  VI 

surface  are  known  as  the  magnetic  "  elements  "  at  that 
point.  They  are  not  constant,  but  are  continually  chang- 
ing. Some  of  these  changes  are  fairly  periodic,  while 
others  are  not.  There  is  a  periodic  change  each  day,  eacli 
month,  and  each  year  ;  and  then  there  is  a  secular  change 
of  such  a  kind  that  the  magnetic  elements  are  gradually 
passing  through  a  cycle  which  may  take  several  hundred 
years  to  complete.  Again,  there  are  sudden  variations, 
which  are  extremely  violent,  and  are  said  to  be  due  to 
"magnetic  storms." 

The  explanation  of  the  magnetism  of  the  earth  is  by  no 
means  understood  at  present ;  and  all  that  can  be  done  is 
to  study  its  field  of  force.  There  is  really  no  such  point 
as  a  magnetic  pole  of  the  earth  ;  but  as  ordinarily  defined, 
the  magnetic  north  pole  of  the  earth  is  that  point  on  the 
northern  hemisphere  where  a  magnetic  needle  would  point 
vertically  down.  The  magnetic  south  pole  of  the  earth 
may  be  defined  in  a  similar  way.  It  must  not  be  thought, 
though,  that,  at  any  other  points  on  the  earth's  surface,  a 
magnetic  needle  points  towards  a  pole  of  the  earth  as  so 
defined ;  a  magnet  always  takes  the  direction  of  the  line 
of  force,  it  does  not  point  towards  any  pole. 


CHAPTEEf  VII 

MAGNETIC  PROPERTIES  OF  STEADY  ELECTRIC   CURRENTS 

As  previously  explained  (Chap.  IV.,  Art.  253),  a  steady 
electric  current  has  an  influence  on  a  magnet  which  may 
be  described  by  saying  that  a  north  pole  tends  to  pass 
around  the  current  in  a  "  right-handed  screw  "  direction, 
while  a  south  pole  tends  to  pass  in  the  opposite  direction. 

273.  Magnetic  Field  around  a  Current.  This  law  may 
then  be  expressed  by  saying  that  around  any  current  there 
is  a  magnetic  field,  and  that  the  lines  of  force  are  closed 
curves  around  the  current  in  a  right-handed  screw  direc- 
tion. Thus,  the  relation  be- 
tween direction  of  current  and 
lines  of  force  is  shown  in 
the  accompanying  figure  :  they 
form  closed  loops  encircling 
each  other.  In  this  case  that 
side  of  the  area  enclosed  by 
the  conductor,  which  is  turned 
towards  the  eye,  has  the  prop- 
erties of  a  magnetic  south 
pole,  because  lines  of  force  FIG.  199. 

are  entering  there ;  while  the 

opposite  side  has  the  properties  of  a  magnetic  north 
pole.  For  this  reason  one  side  of  a  conducting  circuit 
carrying  a  current  is  sometimes  called  its  "  south  side ; " 
and  the  other,  the  "  north  side."  If  the  area  of  the  circuit 
is  made  very  small  by  twisting  the  wires  around  each 
other,  the  magnetic  field  due  to  the  current  almost  com- 
pletely vanishes. 


360 


THEORY  OF  PHYSICS 


[CH.  VII 


Solenoid.  A  special  form  of  conductor  is  a  helix  or 
spiral ;  and,  if  there  is  a  current  in  it,  the  magnetic  lines 
of  force  evidently  enter  one  end,  and  passing  out  the  other 
return  outside  to  form  closed  curves.  When  this  form  of 
a  conductor  carries  a  current,  it  is  sometimes  called  a 


FIG.  200. 

"  solenoid ; "  and  it  has  all  the  external  properties  of  a  bar- 
magnet.  So,  if  a  solenoid  is  suspended  free  to  turn,  it  will 
point  magnetic  north  and  south.  Two  solenoids  carrying 
currents  act  on  each  other  exactly  like  two  bar-magnets. 


FIG.  201. 


274] 


STEADY  ELECTRIC  CURRENTS. 


361 


Further,  it  is  evident  why  a  bar  of  iron  or  steel  becomes 
a  magnet  when  it  is  placed  inside  such  a  solenoid ;  for  the 
lines  of  magnetic  force  come  in  at  one  end  and  leave  at 
the  other,  thus  making  it  a  magnet.  If  the  bar  is  steel, 
it  remains  a  magnet ;  but,  if  it  is  iron,  it  ceases  to  be 
one  when  the  current  stops.  A  solenoid  with  a  soft  iron 
bar  inside  is  sometimes  called  an  "electromagnet."  A 
common  type  of  one  has  its  north  and  south  poles  facing 
each  other,  as  shown  in  Figure  201. 

274.  Electro-magnetic  Force,  Since  electric  currents  pro- 
duce magnetic  fields  around  them,  there  must  be  actions 
between  currents,  and  also  between  a  current  and  a  mag- 
net. (The  action  of  solenoids  is  an  illustration.)  The 
motions  will  always  tend  to  take  place  in  such  a  way  as  to 
produce  a  decrease  in  the  potential  energy  of  the  magnetic 
field ;  and  one  law  has  been  found  which  applies  to  every 
action,  if  a  magnet  may  be  regarded  as  a  solenoid :  the  mo- 
tion always  tends  of  itself  to 
take  place  in  such  a  way  that 
the  field  of  force  passing  from 
the  south  to  the  north  side 
through  a  circuit,  carrying  a 
current,  increases ;  or,  what  is 
the  same  thing,  the  field  of  force 
passing  from  the  north  to  the 
south  side  decreases. 

Thus,  two  parallel  circuits, 
carrying  currents  in  the  same 
direction,  attract  each  other ; 
because  by  coming  nearer  to- 
gether more  lines  of  force  pass 
out  at  the  north  side  of  each. 
This  fact  is  well  shown  in  an  FIG 

experiment   which   consists    in 

allowing  a  current  to  pass  through  a  vertical  spiral  so 
hung  as  to  have  its  lower  end  just  dip  into  a  basin  of 

12* 


362  THEORY   OF  PHYSICS  [CH.  VII 

mercury.  The  current  in  the  spiral  makes  connection 
with  the  cell  through  the  mercury.  As  a  consequence 
of  the  attraction  of  the  parallel  coils  which  have  parallel 
currents,  the  spiral  contracts  and  so  breaks  connection  at 
the  mercury  surface ;  the  current  is  thus  broken,  and  the 
helix  falls ;  it  thus  makes  contact  again  at  the  mercury, 
and  then  the  process  repeats  itself. 

A  solenoid,  with  the  current  as  shown,  has  its  south 
side  at  the  left,  and  its  north  at  the  right.     So,  if  a  bar- 


FIG.  203. 

magnet  is  placed  at  the  right  of  the  solenoid  with  its  south 
pole  towards  the  latter,  there  will  be  attraction  ;  because, 
if  the  two  approach  each  other,  more  lines  of  force  come 
out  the  north  side  of  both  solenoid  and  magnet.  If  either 
current  or  magnet  is  reversed,  there  will  be  repulsion. 

Two  parallel  circuits,  carrying  currents  in  opposite  di- 
rections, will  repel  each  other,  because,  when  they  are 
close  together,  one  almost  neutralizes  the  magnetic  field  of 
the  other ;  and  by  moving  apart  the  field  of  force  coming 
out  the  north  side  of  each  will  be  increased. 

275.  Ampere's  Theory  of  Magnetism.  Ampere  proposed  an 
electric  theory  of  magnetism  which  is  exceedingly  simple 
and  satisfactory  in  many  respects.  Magnetic  substances 
consist  of  molecules  which  are  themselves  magnets  ;  and 
Ampere's  theory  is  simply  that  each  molecule  of  a  mag- 


277]  STEADY  ELECTRIC  CURRENTS  363 

netic  substance  has  in  it  an  electric  current,  which  prob- 
ably flows  in  a  fixed  channel.  If  the  substance  is  not  a 
magnet,  these  molecules  are  standing  at  random  ;  but,  if  a 
bar  of  the  substance  is  made  a  magnet,  the  process  con- 
sists in  the  turning  of  all  the  molecular  currents  until  they 
are  more  or  less  parallel.  This  theory  explains  at  once 
why  a  bar  of  magnetic  material  becomes  a  magnet  when 
placed  inside  a  solenoid,  because  each  molecular  current 
tends  to  turn  and  place  itself  parallel 
to  the  coils  of  the  helix,  so  as  to  have 
as  many  lines  of  force  as  possible  pass 
through  from  its  south  to  its  north  side. 
This  explains,  too,  why  a  bar-magnet 
has  the  same  external  action  as  a  sole- 
noid, because,  considering  the  ends  of 
the  magnet,  lines  of  force  are  passing  pIG.  204. 

in  and  coming  out  through  the  molecu- 
lar currents  just  as  if  there  was  a  helix  wound  around 
the  bar,  and   this  helix  was  carrying   a   current   in  the 
proper  direction. 

276.  Electric  Motors.      The    principle    of    the    ordinary 
electric  motor,  so  common  in  street  cars,  elevators,  etc., 
is   simply  an   application   of    the   general   law   explained 
above.     Coils  of  wire  are  arranged  so  that  they  can  re- 
volve in  the  field  between  two  magnetic  poles.     A  current 
is  passed  from  outside  through  these  coils,  which  then  turn 
so  as  to  have  as  many  lines  of  force  as  possible  pass  out 
from  their  north  sides.     By  an  automatic  device,  the  cur- 
rent is  so  regulated  and  directed  that  each  of  the  coils  is 
tending  to  turn ;  and,  as  the  coils  are  rigidly  fastened  to 
a  shaft,  this  shaft  will  revolve.     By  fastening  gearing  or 
belting  to  this  shaft,  the  motor  may  be  made  to  do  work. 
A  particular  type  of  motor  will  be  explained  later  in  Chapter 
VIII.  Article  290. 

277.  Unit-Current.     There  is,  of  course,  a  connection  be- 
tween the  intensity  of  the  electric  current  and  the  inten- 


364  THEORY  OF  PHYSICS  [CH.  VII 

sity  of  the  magnetic  field  produced  by  it.  It  may  be 
proved  by  experiment  that,  if  a  current  is  passed  around  a 
circuit  of  a  conductor  made  in  the  form 
of  a  circle,  of  radius  r,  the  intensity  of 
the  magnetic  field  at  0,  the  centre  of 
the  circle,  varies  as  the  intensity  of  the 
current,  and  inversely  as  the  radius. 
If  i  is  the  intensity  of  the  current,  and 
f  the  intensity  of  the  magnetic  field  at 

<£ 
FIG.  205.  0,  f  is  proportional  to  - ;  or,  writing  c 

q 

for  a  factor  of  proportionality,  /  =  c  ~ .  /  can  be  meas- 
ured as  previously  explained  (see  Arts.  271  and  269) ; 
so  can  r,  because  it  is  a  distance.  But  the  numerical 
value  of  i  is  entirely  arbitrary,  depending  upon  the 
value  of  c,  which  is  any  constant.  If  it  is  wished,  c  can 
be  made  equal  to  2  TT,  in  which  case 


and  this  equation  fixes  the  numerical  value  of  the  in- 
tensity of  that  current  which,  when  passed  around  a  con- 
ductor bent  in  the  form  of  a  circle  of  radius  r,  will  produce 
the  magnetic  intensity  /  at  the  centre.  This  is  equivalent 
to  giving  a  definition  to  a  current  of  intensity  one,  i.  e.  to 
a  "  unit-current;"  because,  if  /  =  1  and  r  —  2  TT,  then  i  =  1. 
This  means  that  a  unit  current  is  such  that,  if  passed 
around  a  conductor  bent  in  the  form  of  a  circle  whose 
radius  is  2  TT,  it  will  produce  a  magnetic  intensity  1  at  the 
centre.  This  unit  current  is  called  the  "  electro-magnetic 
unit ; "  and  the  system  based  on  it  is  called  the  "  electro- 
magnetic system."  A  unit  quantity  of  electricity  on  this 
system  would  be  the  quantity  carried  by  a  unit  current  in 
one  sec.  (or  by  a  current  of  intensity  2,  in  ^  sec.,  etc.)  It 
has  been  found  by  experiment  that  there  are  almost  ex- 


278]  STEADY  ELECTRIC  CURRENTS  '365 

actly  3  X  1010  electrostatic  units  of  quantity  in  one  electro- 
magnetic unit. 

278.  Tangent  Galvanometer.  If  a  wire  is  wound  so  as  to 
form  n  circular  coils,  of  the  same  radius  r  ;  and,  if  the  helix 
is  so  compressed  that  the  coils  practically  coincide,  the 
magnetic  intensity  at  their  centre,  produced  by  a  current 
of  intensity  i,  will  be  n  times  as  great  as  that  due  to  a 
single  coil.  In  this  case,  then, 


The  direction  of  the  magnetic  force  at  the  centre  is,  of 
course,  perpendicular  to  the  plane  of  the  coils  ;  so  that,  if 
the  coils  are  vertical,  and  if  a  small  magnetic  needle  is 
suspended  at  their  centre,  it  will  tend  to  point  at  right 
angles  to  the  coils.  But,  if  the  coils  themselves  are  in  the 
magnetic  meridian,  the  needle  at  the  centre  is  also  acted 
upon  by  a  moment,  due  to  the  earth's  field,  which  tends  to 
place  it  in  the  plane  of  the  coil.  Consequently,  the  needle 
will  be  deflected,  and  will  come  to  rest  at  some  angle 
where  the  moments  due  to  the  fields  of  the  earth  and  the 
current  balance  each  other. 

Such  an  apparatus  is  called  a  "  tan- 
gent galvanometer,"  for  reasons  which 
will  appear  in  Formula  (2).     It  con- 
sists of  a  wire  wound    in  several  cir- 
cular coils  placed  close  together,  with 
their  planes  in  the  magnetic  meridian, 
and  a  magnetic  needle    at  the  centre 
of   the  coils,  supported  on   a  vertical 
pivot.     The  radius   6f   the  coils   must 
be  immensely  larger  than  the  length 
of   the    needle,  so    that    the    intensity   of    the    magnetic 
force    due    to    the    current    may   be   the    same    at    the 
two  ends  of  the  needle  as  at  its  centre.     Let  the  mag- 
netic moment  of  the  needle  be  Mt  and  when  a  current 


366  THEORY  OF  PHYSICS  [CH.  VII 

passes  through  the  coils,  let  it  come  to  rest  at  an  angle  0 
with  the  magnetic  meridian.  There  are  two  fields  of 
force  acting  on  the  needle,  whose  intensities  are  H,  the 

y      2  TT  n  i\ 
horizontal  intensity  of  the  earth's  field,  and  /  (  = J 

the  intensity  due  to  the  current.     These  two  fields  are  at 


right  angles  to  each  other  ;  so  that  the  two  mechanical  mo- 
ments acting  on  the  magnet  are 

HMw  0  and  fM  cos  0. 

These  two  moments  must  be  equal,  since  the  needle  is  at 
rest;  hence 


HM  sin  6  =  M  cos  0-, 


r 


or  i  =      --  tan  6  .....     (2) 


-  is  constant  for  a  given  instrument,  and  is  called  the 

galvanometer  constant."     Writing 
2  TTU 


the  above  equation  becomes 

*  =  ? tan* <2a) 

Therefore,  if  H,  G,  and  6  can  be  measured,  the  numerical 
value  of  i  is  determined,  and  for  this  reason  the  instru- 


279]  STEADY  ELECTRIC   CURRENTS  367 

ment  is  called  a  "  galvanometer."  In  any  case,  even  if 
H  and  G  are  unknown,  the  values  of  the  intensities  of  two 
currents  may  be  compared  by  finding  the  deflections  pro- 
duced by  them.  Or,  if  i  can  be  measured  by  a  voltameter 
(see  Art.  246),  and  G  and  6  also  measured,  the  value  of  H 
may  be  determined  for  the  place  where  the  galvanometer 
stands. 

This  same  instrument  also  furnishes  a  method  for  the 
comparison  of  the  E.  M.  F.'s  of  two  primary  cells.  For,  by 
Ohm's  law,  i  =  E  /  R  ;  and,  if  the  two  cells,  having  E.  M.  F.'s 
JSi  and  U2,  are  each  allowed  to  produce  a  current  in  turn 
through  the  same  resistance, 


But  the  ratio  ii  /  i2  can  be  measured  by  placing  a  galva- 
nometer in  the  circuit  ;  and  thus  the  ratio  EI  /  Ez  may  be 
determined.  Other  and  better  methods  are  described  in 
laboratory  manuals. 

279.   Electro-magnetic  System,     The  electro-magnetic  unit- 
current  and  unit-quantity  of  electricity  have  been  already 
denned  ;  and  using  them  it 
is  possible  to  define  a  unit 
E.  M.  F.  and  a  unit  resist- 
ance in  this  same  system.    '  FIG  20g 

If,  as  a  current   of   inten- 

sity i  flows  for  t  seconds  through  a  conductor,  the  heat- 
effect  in  a  given  portion  of  that  conductor  from  A  to  £ 
is  H  ergs,  it  has  been  proved  that  H  =  E  it,  if  E  is  the 
E.  M.  F.  between  A  and  B.  But  the  numerical  values  of 
H,  i,  and  t  are  known  ;  so  that  of  E  is  fixed.  This  is 
obviously  equivalent  to  defining  a  unit  E.  M.  F. 

This  same  amount  of  heat-energy  can  be  expressed  in 
terms  of  R,  the  resistance  of  the  conductor  between  A  and 
B.  For,  from  Formula  (11),  Chapter  IV.,  H  =  i2  R  t.  The 
numerical  values  of  H,  i,  and  t  are  known  ;  so  that  of  R  is 
fixed  ;  and  this  is  equivalent  to  defining  a  unit  resistance. 


368  THEORY  OF  PHYSICS  [CH.  VII 

These  electro-magnetic  units  of  E.  M.  F.  and  resistance,  as 
deduced  from  the  unit-current,  are  inconveniently  small ; 
so  certain  multiples  of  these  units  are  ordinarily  used  in 
practical  measurements. 

280.  Practical  Units.  The  practical  unit  of  E.  M.  F.  is 
called  the  "  volt ;  "  and 

1  volt  =  108  electro-magnetic  units. 

The  practical  unit  of  resistance  is  called  the  "  ohm ;  "  and 
1  ohm  =  109  electro-magnetic  units. 

The  practical  unit  of  intensity  of  current  is  called  the 
"  ampere  ; "  and 

1  ampere  =  10"1  electro-magnetic  unit. 

(Hence  an  E.  M.  F.  of  1  volt  applied  at  the  ends  of  a  con- 
ductor of  1  ohm  resistance  produces  a  current  of  1  ampere.) 
The  practical  unit  of  quantity  is  called  the  "coulomb;" 
and,  of  course, 

1  coulomb  =  10~ l  electro-magnetic  unit. 

Even  these  units,  although  of  convenient  size,  are  not  well 
adapted  for  ordinary  daily  use.  For  instance,  in  order  to 
find  the  E.  M.  F.  between  two  points  of  a  conductor,  it 
would  be  most  inconvenient  to  measure  the  heat-effect  pro- 
duced as  the  current  passes  ;  although  it  might  be  done  in 
some  cases.  Similarly,  the  resistance  could  not  be  con- 
veniently measured.  But  methods  are  known  by  which 
the  ratio  of  two  E.  M.  F.'S  or  the  ratio  of  two  resistances 
may  be  most  accurately  determined.  Therefore,  with  the 
greatest  exactness,  the  value  of  a  definite  standard  E.  M.  F. 
has  been  measured  in  terms  of  volts ;  and  the  value  of  a 
definite  standard  resistance  has  been  measured  in  terms  of 
ohms.  These  standards  can  be  compared  with  the  un- 
known quantities  ;  and  so  values  of  the  latter  may  be 
determined. 

The  E.  M.  F.  between  the  poles  of  a  Clark  cell,  made  in  a 


281]  ELECTRIC   CURRENTS  369 

particular  way,  is  1.434  volts  if  the  temperature  is  15°  C. 
Specifications  may  be  easily  obtained  for  the  construction 
of  the  cell;  and  the  rate  of  change  of  its  E.  M.  F.  with  the 
temperature  is  known  ;  so  there  is  no  difficulty  in  the  use 
of  this  standard  E.  M.  F. 

The  resistance  of  a  column  of  mercury  106.3  cm.  long  of 
uniform  cross-section,  and  containing  14.4521  grams  is  1 
ohm  at  0°  C.  Mercury  is  easily  obtained  in  a  pure  con- 
dition ;  and  so  a  column  of  it  in  a  glass  tube  of  the  proper 
size  may  serve  for  a  standard  resistance. 

As  stated  before,  in  Table  XIV.,  when  an  electro-mag- 
netic unit  quantity  of  electricity  is  carried  through  a  solu- 
tion of  a  silver  salt,  0.01118  grams  of  silver  are  deposited. 
This  is  absolutely  true  only  if  the  solution  is  one  of  silver 
nitrate  in  water,  special  precautions  being  observed.  A 
coulomb  will  therefore  deposit  0.001118  grams  of  silver; 
or,  what  is  the  same  thing,  96,540  coulombs  will  deposit 
an  amount  of  any  substance  equal  to  its  chemical  equiva- 
lent. By  means  of  a  knowledge  of  the  above  figures  for 
a  coulomb  and  a  silver  salt,  the  intensity  of  any  current 
may  be  measured  with  any  kind  of  voltameter. 

281.  Energy  of  a  Magnetic  Field.  Since  a  magnet  has 
all  the  properties  of  an  electric  current,  in  discussing  the 
energy  of  a  magnetic  field  it  will  be  necessary  to  speak 
only  of  currents.  There  is  no  magnetic  field  around  a  con- 
ductor which  is  not  carrying  a  current ;  but,  as  a  current 
is  started,  a  magnetic  field  is  produced,  and  work  is  re- 
quired to  do  this.  This  is  obvious,  because  changes  are 
produced  in  the  surrounding  ether  and  matter.  Conse- 
quently, a  definite  amount  of  work  is  required  to  start  a 
current,  to  raise  it  from  zero  up  to  its  full  value ;  and 
this  energy  is  undoubtedly  present  as  kinetic  energy  in 
the  ether  and  the  connected  matter.  After  the  current  is 
started,  energy  is  required  to  keep  it  going,  owing  to  the 
work  necessary  to  overcome  the  resistance  of  the  conductor, 
this  energy  being  spent  in  heat-effects  in  the  conductor. 


370  THEORY  OF   PHYSICS  [CH.  VII 

So  every  current  (and  magnet)  must  be  considered  as 
associated  with  a  definite  amount  of  kinetic  energy  in  the 
surrounding  medium.  The  amount  of  this  energy  depends, 
of  course,  on  the  extent  and  intensity  of  the  magnetic  field 
produced  by  the  current,  that  is,  on  the  form  of  the  con- 
ducting circuit  and  on  the  intensity  of  the  current. 

If  there  are  two  currents  near  each  other  (or  two  mag- 
nets), there  is  an  extra  amount  of  energy  in  the  field,  as  is 
shown  by  the  fact  that  one  current  has  an  action  on 
another.  This  energy  is  potential,  and  depends  upon  how 
much  work  has  been  required  to  bring  these  two  currents 
near  each  other,  if  originally  they  were  an  infinite  distance 
apart.  If  work  was  done  by  external  forces  in  bringing 
the  currents  into  position,  this  energy  is  added  to  that 
of  the  two  currents  ;  while,  if  the  two  currents  "  attract " 
each  other,  the  energy  is  subtracted  from  that  of  the 
currents. 


CHAPTER  VIII 

INDUCED  CURRENTS 

IF  a  conducting  circuit,  e.g.  a  wire  bent  into  a  closed 
curve*,  is  placed  in  a  magnetic  field,  there  will  in  general 
be  a  certain  field  of  force  passing  through  the  area  bounded 
by  the  circuit.  This  is  true  whether  the  circuit  carries  a 
current  itself  or  not.  Faraday  discovered  that,  if  in  any 
way  this  field  of  force  passing  through  the  circuit  ia 
changed,  there  is  instantly  a  current  produced  in  the  con- 
ducting circuit,  or,  if  there  is  a  current  there  already,  it 
will  be  changed.  Faraday  called  these  currents  which  are 
thus  produced  by  changing  the  field  of  magnetic  force 
"  induced  currents." 

282.  Properties  of  Induced  Currents.  These  induced  cur- 
rents last  only  as  long  as  the  field  of  force  through  the 
circuit  is  being  changed;  and  it  is  proved  by  experiment 
that  they  are  always  in  such  a  direction  as  by  their  own 
action  to  oppose  or  neutralize  the  change  in  the  field  by 
which  they  are  produced;  the  amount  of  the  "induced 
E.  M.  F.,"  to  which  they  may  be  attributed,  may  be  proved 
to  vary  directly  as  the  amount  of  the  change  in  the  field  of 
force  through  the  circuit,  and  inversely  as  the  time  taken 
for  the  change  to  be  produced.  Further,  if  there  are  n 
turns  of  wire  in  the  circuit,  the  induced  E.  M.  F.  will  clearly 
be  n  times  as  great  as  it  would  be  for  a  single  coil. 

Energy  is  required  for  the  production  of  these  induced 
currents ;  and,  by  considering  certain  simple  illustrations, 
the  sources  of  the  energy  will  be  clearly  understood. 


V 


372  THEORY  OF  PHYSICS  [CH.  VIII 

SPECIAL   CASES 

283.  1.  A  Current  and  a  Magnet.  Let  the  arrangement 
be  as  shown ;  a  magnet  turned  with  its  south  pole  towards 
the  south  side  of  the  current.  There  is  a  definite  field 

of  force  passing  inside  the  cir- 
cuit, due  to  the  current  and 
to  the  magnet.  There  will  be 
an  induced  current  if  this  is 
1  '  in  any  way  changed.  Let  the 

circuit  be  kept  fixed,  and  let 
the  magnet  be  moved  towards 
it.     Work  is  required  as  long 
FIG.  209.  as  there  is  motion ;   and  en- 

ergy, therefore,  leaves  the  ex- 
ternal body  which  produces  the  motion,  and  is  available 
for  the  induced  current  which  flows  while  the  magnet  is 
being  moved. 

The  effect  on  the  field  of  force  is  to  decrease  the  field 
of  force  passing  through  the  circuit  from  the  south  side  to 
the  north ;  and  the  induced  current  will  be  in  such  a 
direction  as  to  increase  this  field ;  that  is,  it  will  be  in 
the  same  direction  as  the  original  current.  As  soon  as  the 
motion  of  the  magnet  ceases,  the  supply  of  energy  ceases  ; 
and  the  induced  current  dies  out  rapidly,  its  energy  being 
spent  in  heating  the  conducting  circuit. 

This  induced  current  presents  its  south  side  towards 
the  approaching  magnet,  arid  tends  to  repel  it;  and  the 
phenomenon  may  be  described  by  saying  that,  when  the 
magnet  is  brought  near,  the  induced  current  is  in  such 
a  direction  as  to  repel  it,  that  is  to  tend  to  prevent  the 
motion. 

Again,  suppose  that  both  magnet  and  circuit  are  free 
to  move.  They  will  separate,  owing  to  the  tendency  of 
the  potential  energy  to  become  less  ;  but  this  changes 
the  field  of  force.  As  they  move  apart,  the  field  of  force 


284]  INDUCED   CURRENTS  373 

passing  through  from  the  south  to  the  north  side  of  the 
circuit  is  increasing  ;  so  the  induced  current  must  be  in 
such  a  direction  as  to  decrease  this  field ;  that  is,  it  is 
in  the  opposite  direction  to  the  existing  current,  and  the 
immediate  result  is  a  decrease  in  the  actual  current. 

Work  is  required  to  produce  the  motion  of  separation  of 
the  circuit  and  magnet ;  for  work  is  always  required  to  set 
any  body  in  motion.  The  only  source  of  energy  is  the 
cell  which  furnishes  the  current ;  and,  if  some  of  its 
energy  is  used  up  in  producing  motion,  less  than  usual 
is  available  for  the  current ;  and  consequently  the  current 
decreases.  As  soon  as  the  motion  ceases,  the  current  re- 
turns to  its  previous  value. 

The  induced  current  was  in  such  a  direction  as  to  have 
its  north  side  towards  the  magnet ;  that  is,  it  tended  to 
hinder  the  motion  of  separation.  The  phenomenon  may, 
then,  be  described  by  saying  that  the  induced  current  is  in 
such  a  direction  as  to  tend  to  oppose  the  change  in  posi- 
tion of  the  circuit  and  magnet.  This  law  is,  in  fact,  a 
general  one,  and  applies  to  all  changes  in  position  of  cur- 
rents and  magnets. 

It  is  at  once  evident  how  these  same  explanations  may 
be  extended  to  the  motions  of  currents  with  reference  to 
each  other. 

284.    2.  Conducting-Circuit  and  Magnet.     If  a  con  ducting- 
circuit,  which  has  in  it  no  permanent  current,  is  placed 
near  a  magnet,  as  shown,  there 
will  be  a  field  of  force  passing 
inside   the    circuit.      If   this    is 
changed  in  any  way,  there  will 
be  an  induced  current.     Let  the 
circuit    be    fixed,    and    let    the     \          / 
magnet  be  moved  nearer.     The 
change    in    the    field    of    force  FlG  2io. 

through   the    circuit   is    to    in- 
crease the  field  coming  through  the  circuit  from  the  side 


374  THEORY  OF  PHYSICS  [CH.  VIII 

away  from  the  magnet  to  the  side  towards  it ;  conse- 
quently the  induced  current  is  in  such  a  direction  as  to 
decrease  this  field ;  that  is,  it  is  such  as  to  make  its  south 
side  face  the  approaching  magnet.  (Here  again  the  in- 
duced current  is  in  such  a  direction  as  to  hinder  the 
change.)  Work  is  done  by  the  force  which  brings  the 
magnet  up,  owing  to  this  opposing  current ;  and  so  the  en- 
ergy is  accounted  for. 

Similarly,  if  the  magnet  is  moved  away,  the  induced 
current  is  in  such  a  direction  as  to  have  its  north  side 
towards  the  retreating  south  pole  ;  and  again  the  motion  is 
hindered.  In  short,  whenever  the  relative  positions  of  a 
magnet  and  a  conducting  circuit  are  altered,  the  induced 
current  is  in  such  a  direction  as  to  oppose  the  change. 
The  energy  of  the  currents  is,  of  course,  spent  in  heating 
the  conductor. 

285.  A  well-known  illustration  of  these  laws  is  "  Arago's 
disc,"  which  is  a  circular  disc  of  copper  (or  any  conductor) 
so  arranged  as  to  be  rapidly  revolved  in  its  own  plane  over 
(or  under)  a  pivoted  magnet.  Currents  are,  of  course,  in- 
duced in  the  copper  disc ;  and  these  currents  are  in  such 
directions  as  to  tend  to  prevent  the  motion.  Consequently, 
those  portions  of  the  disc  which  are  going  away  from 
either  pole  of  the  magnet  tend  to  attract  it  after  them  so 
that  they  would  not  be  separated ;  portions  of  the  disc 
approaching  the  pole  tend  to  repel  it.  So,  as  the  disc 
revolves,  the  magnet  tends  to  follow  it,  turning  in  the 
same  direction.  The  disc,  in  its  turn,  becomes  hot,  as  a 
result  of  the  currents  ;  and  the  entire  energy  is,  of  course, 
furnished  by  the  power  which  turns  the  disc. 

An  exactly  similar  phenomenon  is  observed  when  a  cop- 
per disc  is  pushed  between  the  poles  of  a  powerful  electro- 
magnet. In  this  case,  also,  the  induced  currents  in  the 
copper  are  such  as  to  oppose  the  motion  ;  and  work  is 
required  to  move  the  disc.  If  it  is  moved  repeatedly 
back  and  forward,  its  temperature  will  soon  rise. 


286] 


INDUCED  CURRENTS 


375 


286.  Another  illustration  of  currents  in  conducting  cir- 
cuits induced  by  variations  in  the  field  of  force  is  afforded 
by  the  so-called  "  earth-inductor."  This  is  a  circular  frame- 
work around  which  is  wound  a  wire  in  several  turns,  and 
which  is  so  supported  as  to  permit  rotation  around  an  axis 
lying  in  the  plane  of  the  coils,  and  passing  through  the 
centre.  Mechanical  stops 
are  so  arranged  as  to  per- 
mit it  to  turn  only  180°, 
and  the  ends  of  the  wire 
are  joined  to  a  galvanom- 
eter. The  apparatus  is 
used  in  this  way  :  it  is  so 
placed  that  the  axis  of 
rotation  is  vertical,  and 
the  plane  of  the  coils  is 
adjusted  exactly  at  right 
angles  to  the  magnetic  me- 


r  T    T 


FIG.  211. 


ridian  ;  it  is  then  turned 
rapidly  through  180°. 
When  the  coils  are  in  their  first  position,  the  field  of 
force  through  them  is  due  to  the  horizontal  component 
of  the  earth's  field ;  and,  when  they  are  turned  90°,  there 
is  no  force  at  all  through  them;  but,  when  turned  90° 
farther,  the  horizontal  field  of  the  earth  again  passes 
through  them,  but  from  the  opposite  side.  So,  in  turning 
through  180°,  the  change  in  the  field  of  force  equals  twice 
the  field  through  the  coils  when  they  are  in  their  first 
position.  There  will,  of  course,  be  an  induced  current; 
and  this  will  be  shown  by  the  galvanometer,  which  is 
joined  to  the  coils  of  wire.  It  is  possible  to  prove  that, 
if  the  needle  of  a  galvanometer  swings  very  slowly  (1  vi- 
bration in  about  12  seconds),  the  extent  of  the  sudden 
fling  of  the  needle,  which  is  produced  by  an  induced  cur- 
rent, is  proportional  to  the  induced  quantity.  (Such  a 
galvanometer  is  called  a  "  ballistic  galvanometer,"  and  may 


376  THEOKY  OF  PHYSICS  [CH.  VIII 

be  used  to  measure  any  sudden  flow  of  electricity.  Strictly 
speaking,  the  quantity  is  proportional  to  the  sine  of  half 
the  angular  fling  of  the  needle ;  but,  if  the  angle  is  small, 
the  sine  of  an  angle  practically  equals  the  angle  itself.) 
The  induced  E.  M.  F.,  and  therefore  the  intensity  of  the 
induced  current,  is,  as  noted  above,  proportional  to  the 
change  in  the  field  of  force,  and  inversely  proportional  to 
the  time  taken.  But  the  induced  quantity  equals  the 
intensity  of  the  current  multiplied  by  the  time ;  and  so 
the  quantity  is  proportional  simply  to  the  change  in  the 
field  of  force.  Consequently,  in  the  experiment  just  dis- 
cussed, the  fling  of  the  galvanometer  needle  is  proportional 
to  the  horizonal  intensity  of  the  earth's  field.  Let  the 
fling  be  through  the  angle  ai,  then  H  equals  some  constant 
times  0,1,  or  H  =  c  ax. 

Now  place  the  coils  of  wire  so  that  they  are  horizontal, 
and  rotate  them  through  180°  about  the  horizontal  axis. 
There  will  be  an  induced 'current,  depending  upon  the 
amount  of  the  vertical  component  of  the  earth's  field. 
Let  the  fling  of  the  galvanometer  be  a2  Then,  as  above, 
V  =  c  a2, 

and,  hence,  -==.  =  — (1) 

a.       a\ 

Thus,  VI H  can  be  measured ;  but,  by  Article  272,  Formula 
(1  a),  this  ratio  is  the  tangent  of  the  dip ;  so  the  dip  may 
be  determined. 

By  means  of  an  earth-inductor  and  ballistic-galvanometer 
the  intensity  of  any  magnetic  field  of  force,  e.  g.  that  be- 
tween the  poles  of  a  dynamo,  may  be  compared  with  that 
due  to  the  horizontal  component  of  the  earth's  field.  There- 
fore, if  H  is  measured  absolutely,  the  intensity  of  any  other 
magnetic  field  may  be  at  once  determined. 

287.  3.  A  Varying  Current  in  a  Conductor.  If  there  is  a 
current  in  a  conductor,  there  is  of  course  a  certain  mag- 
netic field  around  it ;  so  that,  if  this  circuit  is  broken,  the 


288]  INDUCED  CURRENTS  377 

field  tends  to  disappear.  But,  as  it  tends  to  decrease,  there 
must  be  an  induced  current  tending  to  increase  it,  or 
rather  to  keep  it  constant ;  and  consequently  there  will  be 
an  induced  current  in  the  same  direction  as  the  original 
current.  This  is  made  evident  by  the  spark  seen  when  a 
circuit  which  carries  a  current  is  broken.  As  explained 
in  the  last  chapter,  Article  281,  a  current  is  surrounded  by 
a  magnetic  field  which  contains  energy ;  the  moment  the 
circuit  is  broken,  this  energy  returns  to  the  conductor 
which  carried  the  current;  and  the  current  really  con- 
tinues for  a  short  time  after  the  E.  M.  F.  of  the  cell  is 
removed.  This  induced  current  is  sometimes  called  the 
'•'extra-current  on  breaking." 

Similarly,  when  a  current  is  being  started,  the  field  of 
force  is  increasing ;  and  there  will  be  an  induced  current 
in  such  a  direction  as  to  oppose  the  change,  that  is,  oppo- 
site to  the  main  current.  The  effect  is  that  it  takes  some 
time  to  make  a  current  reach  its  full  value.  In  other 
words,  energy  is  being  given  the  surrounding  medium, 
and,  owing  to  its  inertia,  time  is  necessary  for  a  steady 
state  to  be  reached.  This  induced  opposition-current  is 
sometimes  called  the  "  extra-current  on  making." 

(By  way  of  analogy,  compare  the  fact  that,  when  a  rail- 
way train  starts  from  rest,  some  time  elapses  before  it 
attains  its  full  speed,  work  being  done  in  producing  kinetic 
energy.  After  this  speed  is  reached,  work  is  done  only  in 
overcoming  frictional  resistances.) 

288.  4.  A  Varying  Current  near  a  Conducting  Circuit. 
This  case  is  in  principle  exactly  the  same  as  cases  1  and 
2.  In  them  the  field  of  force  inside  a  conducting  circuit 
was  altered  by  changing  the  relative  positions  of  the  cir- 
cuit and  a  magnet  (or  solenoid).  This  field  may  be 
equally  well  changed  by  altering  the  intensity  of  the  cur- 
rent in  a  solenoid  near  by.  Some  of  the  lines  of  force  due 
to  the  solenoid  pass  through  the  conducting  circuit,  in 
general ;  and,  if  the  intensity  of  the  current  in  the  solenoid 


378  THEORY  OF  PHYSICS  [CH.  VIII 

changes,  the  field  of  force  changes,  and  there  will  be  cor- 
responding induced  currents  in  the  conducting  circuit. 

Many  instruments  have  been  devised  to  make  use  of 
induced  currents ;  and  a  few  will  now  be  discussed. 

289.  Transformer,  or  Induction  Coil.  An  induction  coil, 
or  transformer,  is  an  instrument  designed  to  receive  a  cur- 
rent with  a  small  E.  M.  F.,  and  to  furnish  one  with  a  large 

E.  M.  F. ;  or  vice  versa. 
The  principle  is  exceed- 
ingly simple.  Two  coils 
of  wire  are  so  wound  as 
to  have  the  same  mag- 
netic field  when  a  cur- 
rent passes  in  either. 
This  may  be  secured  by 

FIG  2^  '  having    the    two    coils 

wound  around  the  same 

soft  iron  ring,  even  at  different  portions  of  it ;  or  by  hav- 
ing one  coil  wound  inside  the  other.  One  coil  consists, 
in  general,  of  a  great  number  of  turns  of  fine  wire ;  the 
other,  of  a  few  turns  of  coarse  wire. 

When  a  current  is  passed  through  the  coarse  wire,  even 
if  there  is  a  small  E.  M.  F.,  the  intensity  will  be  great 
because  the  resistance  is  small,  and  so  there  will  be  a 
strong  field  of  magnetic  force,  especially  if  there  is  a  core 
of  soft  iron  inside  the  coils.  This  field  of  force  is  inside 
the  coil  of  fine  wire,  too ;  and,  if  now  by  any  means  the 
current  in  the  coarse  wire  can  be  either  reversed  or  broken, 
there  will  be  a  great  change  in  the  field  of  force,  and  a 
current  will  be  induced  in  the  coil  of  fine  wire.  The  in- 
duced E.  M.  F.  depends  upon  the  rate  of  change  of  the  field 
of  force,  and  also  upon  the  number  of  turns  of  the  wire ; 
because  in  each  turn  there  will  be  the  same  E.  M.  F.  in- 
duced, and  these  are  all  superimposed  so  that  the  E.  M.  F. 
at  the  ends  of  the  coil  varies  directly  as  the  number  of 
turns  of  wire.  The  current  in  the  coarse  wire  may  be 


289]  INDUCED  CURRENTS  379 

reversed  (by  an  "alternating  dynamo")  or  broken  and 
made  again  at  very  frequent  intervals ;  and  so  the  induced 
E.  M.  F.  in  the  fine  wire  will  be  extremely  great.  Conse- 
quently, when  used  in  this  way,  a  small  E.  M.  F.  and  large 
current  give  rise  to  a  large  E.  M.  F.  and  a  correspondingly 
small  current,  because  the  energy  of  a  current  varies  as 
the  product  of  the  E.  M.  F.  and*  the  current;  and  the 
energy  delivered  by  the  apparatus  cannot  exceed  that 
furnished  it. 

Conversely,  if  a  very  large  E.  M.  F.  is  applied  to  the  coil 
of  fine  wire,  it  will  produce  only  a  small  current ;  but  if 
this  is  reversed,  or  broken  and  made  again,  at  regular 
intervals,  there  will  be  a  large  current  with  a  small  E.  M.  F. 
produced  in  the  coil  of  coarse  wire. 

To  reverse  the  current  in  the  coil,  an  alternating  dynamo 
is  used ;  that  is,  a  machine  which  produces  a  current  first 
in  one  direction,  then  in  the  other ;  but  simply  to  make 
and  break  an  ordinary  direct  current,  any  automatic  device 
may  be  used.  The  simplest  form  is  shown  in  Fig.  213 
which  represents  an  ordinary  "  induction  coil."  The  soft 
iron  core  of  the  two  coaxial  coils  extends  a  short  distance 
beyond  the  coils ;  and  quite  close  to  its  end  is  placed  a 
piece  of  soft  iron  which  is  fastened  on  the  end  of  a  stiff 
spring.  When  this  spring  is  in  its  natural  position,  it 
completes  the  circuit  from  the  battery  through  one  of  the 
coils  of  wire ;  but,  when  the  soft  iron  core  is  magnetized 
by  the  current,  it  attracts  the  soft  iron  on  the  end  of  the 
spring,  and  so  breaks  the  circuit.  When  the  circuit  is 
broken,  the  iron  core  ceases  to  be  a  magnet,  the  spring  flies 
back  to  its  previous  position,  contact  is  made,  and  the  cur- 
rent again  flows ;  and  the  process  is  repeated  indefinitely. 
As  the  field  of  force  through  the  core  is  increased  and  then 
decreased  in  succession,  it  might  be  expected  that  the 
induced  current  in  the  other  coil  would  be  first  in  one 
direction  and  then  in  the  other.  This,  in  fact,  is  observed 
unless  special  precautions  are  taken.  In  an  induction-coil, 


380 


THEORY  OF  PHYSICS 


[CH.  VIII 


a  condenser  is  commonly  inserted,  as  shown,  in  the  battery 
circuit ;  and  this  will  in  general  so  weaken  the  E.  M.  F.  on 
making  contact  that  it  produces  comparatively  little  effect. 
This  is  done  because  an  induction  coil  is  ordinarily  used  to 
produce  spark-discharges  in  one  direction  between  the  two 


FIG.  213. 

terminals  of  the  coil  in  which  the  currents  are  induced  ;  and 
unless  the  E.  M.  F.  exceeds  a  certain  limit,  sparks  cannot 
pass.  It  is  by  means  of  induction-coils  in  general,  that 
sparks  and  discharges  through  gases  are  studied.  (See 
Art.  250.) 

The  iron  core  of  the  coils  of  wire  is  always  made  of 
small  iron  wires,  carefully  insulated,  placed  side  by  side. 
If  the  core  was  solid,  there  would  be  currents  induced 
around  the  iron  core  itself,  as  a  result  of  the  varying 
current ;  and  a  great  deal  of  energy  would  be  wasted  in 


290] 


INDUCED   CURRENTS 


381 


heating  the  iron  But,  if  it  is  made  of  wires,  insulated  from 
each  other,  the  resistance  is  so  great  from  one  wire  to  an- 
other that  a  current  cannot  pass. 

.  290.  Dynamos.  The  simplest  case  of  a  so-called  dynamo 
is  that  of  the  "  Gramme-ring "  type.  It  consists  of  two 
parts,  —  the  magnet  and  the  armature.  The  magnet  is  one 
so  made  that  the  north  and  the  south  poles  come  opposite 


FIG.  214. 

each  other  (as  in  an  ordinary  horse-shoe  magnet  or  elec- 
tro-magnet) ;  and  it  may  be  a  permanent  steel  one  or  an 
electro-magnet  which  is  magnetized  by  an  electric  cur- 
rent. The  armature  consists  of  a  soft  iron  ring  which  is 
made  up  of  insulated  iron  wires  bent  into  circles,  and 
around  which  is  wound  a  continuous  copper  wire  carefully 
insulated  from  the  iron.  The  armature  is  rigidly  fastened 
to  a  shaft  perpendicular  to  its  plane ;  and  the  shaft  is 
placed  perpendicular  to  the  magnetic  field  of  force  between 
the  poles  of  the  magnet.  If  the  shaft  is  revolved,  cur- 


382  THEORY  OF  PHYSICS  [CH.  VIII 

rents  will,  of  course,  be  induced  in  the  coils  of  wire  wound 
around  the  ring,  because  the  field  of  magnetic  force 
through  them  is  constantly  changing  as  the  ring  revolves. 
On  the  shaft  of  the  armature  is  fastened  what  is  called 
the  "  commutator,"  which  consists  of  metal  strips  or 
"  bars  "  along  the  shaft,  each  insulated  from  its  neighbors  ; 
and  resting  across  these  bars  are  two  so-called  "  brushes," 
which  are  metal  strips,  one  on  one  side  of  the  commutator, 
the  other  diametrically  opposite,  so  arranged  as  to  touch 
opposite  bars  of  the  commutator  at  the  same  instant. 
These  brushes  are  held  stationary,  as  the  commutator  re- 
volves ;  and  they  are  joined  by  a  conductor,  through  which 
a  current  is  desired.  Each  of  the  bars  of  the  commutator 
is  joined  by  a  wire  to  different  points  of  the  wire  which  is 
wound  around  the  iron  ring.  Consequently,  as  the  arma- 
ture is  turned  by  means  of  the  shaft,  the  brushes  are 
always  joined  to  those  portions  of  the  wire  around  the 
iron  ring  which  occupy  in  turn  the  same  positions  in  the 
magnetic  field. 

The  lines  of  force  from  the  north  pole  of  the  magnet 
pass  into  the  iron  ring  and  around  through  the  ring  to  the 
side  opposite  the  south  pole,  where  they  pass  out  and 
cross  the  air-gap  to  the  pole.  They  do  not  pass  across  the 
ring,  but  are  divided,  as  it  were,  into  two  sections  which 
crowd  through  at  the  top  and  bottom  of  the  ring.  (Com- 
pare Fig.  194,  Art.  267.)  Consequently,  as  the  armature 
revolves,  turns  of  wire  which  are  horizontal  have  no  field 
of  force  through  them  ;  but,  as  they  reach  the  top  or  bot- 
tom and  so  are  placed  vertical,  there  is  a  strong  field 
through  them.  Currents  are,  therefore,  induced  in  these 
coils;  and  their  directions  are  easily  deduced.  If  the 
dynamo  is  as  shown,  and  the  armature  is  being  turned  as 
shown,  it  is  seen  that  the  currents  in  all  the  coils  on  the 
ascending  half  of  the  ring  are  downward,  while  those  in 
the  other  half  are  also  downward.  So,  if  the  brushes  are 
connected  with  those  two  bars  of  the  commutator  which 


291] 


INDUCED   CURRENTS 


383 


are  joined  to  the  top  and  bottom  coils  of  the  armature, 
there  will  be  a  current  produced  which  will  flow  from  one 
brush  around  to  the  other ;  then  to  the  top  coil,  where  it 
will  divide  into  two  branches  which  meet  at  the  bottom 
coil ;  and  then  back  to  the  other  brush.  This  process  is 
perfectly  continuous,  and  a  steady  current  will  be  produced. 

In  practice,  the  brushes  are  not  connected  with  the  coils 
at  top  and  bottom,  but  with  those  a  little  farther  in  ad- 
vance, in  the  direction  of  rotation.  This  is  because,  as  the 
current  is  produced  in  the  armature,  it  produces  a  mag- 
netic field  which  so  influences  the  field  due  to  the  mag- 
nets, that  the  coil  where  the  current  tends  to  branch  is  no 
longer  exactly  at  the  top,  but  is  a  little  in  advance. 

If,  instead  of  driving  the  shaft  of  this  armature  by 
means  of  some  external  power  and  so  producing  a  current, 
a  current  is  sent  through  the  armature  from  some  other 
dynamo  or  a  battery,  the  armature  will  revolve  and  the 
shaft  can  be  used  to  furnish  power.  This  is,  of  course,  the 
principle  of  the  motor,  as  already  mentioned. 

291.  Telephone.  The  ordinary  telephone  consists  of  a 
long  steel  magnet  having  a  thin  coil  of  wire  wound 


FIG.  215. 


around  one  end,  and  a  soft  iron  diaphragm  held  a  short 
distance  away  from  this  end.  The  soft  iron  plate  becomes 
a  magnet  under  the  influence  of  the  steel  magnet ;  and,  if 
it  is  made  to  approach  the  latter,  it  will  alter  the  field  of 


384 


THEORY  OF  PHYSICS 


[CH.  VIII 


force  passing  through  the  coil  of  wire,  and  there  will  be  an 
induced  current  in  a  certain  direction.  Similarly,  if  the 
plate  is  moved  away,  there  will  be  an  induced  current  in 
the  opposite  direction.  Consequently,  if  sounds  are  made 
near  the  plate,  it  will  be  moved  back  and  forward  by  the 
sound-waves ;  and  corresponding  induced  currents  will  be 
produced. 

If  a  current  is  sent  from  outside  through  the  coil  of 
wire  at  the  end  of  the  magnet,  it  will  either  strengthen  or 
weaken  the  magnetic  field ;  and  the  soft  iron  diaphragm 
will  be  either  attracted  or  repelled.  If  the  current  was 
in  the  opposite  direction,  the  opposite  motion  would  be  ob- 
served. Consequently,  if  a  current  which  is  being  rapidly 
reversed  is  sent  through  the  coil,  the  plate  will  have  cor- 
responding motions  back  and  forward. 

If,  then,  two  telephones  are  joined  so  that  the  wire  from 
one  coil  is  connected  in  series  with  that  from  the  other, 
forming  a  closed  circuit,  any  motion  of  the  plate  of  one 
will  produce  a  corresponding  motion  in  that  of  the  other ; 
and  any  sound  made  near  one  will  produce  corresponding 
vibrations  in  the  other,  which  will  in  turn  send  out 
sound-waves. 

292.  Microphone.  A 
microphone  consists,  in 
principle,  of  a  primary 
cell  joined  in  series  with 
a  "  bad-contact ;  "  that 
is,  a  junction  between 
two  conductors  where 
one  rests  rather  loosely 
against  the  other.  The 
resistance  of  this  bad- 
FIG.  216.  contact  will  vary  greatly 

according  to  the  pressure 

of  the  two  loose  conductors ;  and,  if  this  pressure  is  changed 
intermittently  by  a  series  of  jars  or  shocks,  there  will  be 


nnnnnr 


293]  INDUCED   CURRENTS  385 

corresponding  fluctuations  in  the  current  flowing  from  the 
cell.  So,  if  a  telephone  is  joined  in  series  with  the  bad- 
contact,  any  alteration  of  the  contact  will  produce  a  cor- 
responding motion  of  the  telephone  diaphragm.  In  the 
ordinary  microphone  used  for  speaking  purposes,  the  bad- 
contact  is  between  a  plate  of  metal  and  a  small  metallic 
button  back  of  it ;  and,  if  a  sound  is  made  near  the  plate, 
there  are  alterations  in  the  contact,  and  these  produce  a 
sound  in  the  telephone  which  is  in  series  with  it. 

293.  Diamagnetism.  There  is  a  theory  of  diamagnetism, 
which  was  proposed  by  Weber,  and  which  is  based  upon 
the  properties  of  induced  currents.  The  fundamental 
property  of  a  diamagnetic  substance  is  that  if  placed  near 
a  magnet  it  will  be  repelled.  In  Weber's  theory  it  is 
assumed  that  the  molecules  of  a  diamagnetic  substance  are 
perfect  conductors,  but  that  they  do  not  have  any  currents 
flowing  in  them,  or  at  least  that  they  are  extremely  feeble. 
If  such  a  substance  is  brought  near  the  north  pole  of  a 
magnet,  there  will  be  an  increase  in  the  number  of  lines 
entering  through  the  molecules  of  the  substance.  Conse- 
quently there  will  be  an  induced  current  in  each  so  as  to 
send  lines  of  force  out ;  that  is,  these  currents  will  be  in 
such  directions  as  to  have  their  north  sides  outward.  There 
will,  therefore,  be  repulsion  between  the  magnet  and  the 
diamagnetic  substance.  If  the  latter  is  taken  away  from 
the  neighborhood  of  the  magnet,  currents  will  be  induced 
in  the  opposite  direction,  and  the  molecules  will  return  to 
their  previous  condition. 


CHAPTER  IX 
GENERAL  PROPERTIES  OF  A  MAGNETIC  FIELD 

294,  THERE  is  a  great  deal  of  evidence  for  believing  that 
in  a  magnetic  field  there  is  a  rotation  or  spinning  of  the 
minute  portions  of  the  ether  and  matter.     This  spinning 
is  not  motion  of  large  portions  of  matter,  but  is  just  as  if 
a  series  of  molecules  were  strung  on  a  line,  and  this  line 
were  rapidly  rotated  around  itself  as  an  axis.    It  is  thought 
that  there  is  some  motion  like  this  associated  with  every 
magnetic  line  of  force.     That  rotation  of  some  kind  is  con- 
nected with  magnetic  fields  is  shown  by  two  experiments. 

295,  Rotation  of  Plane  of  Polarization.     It  will  be  shown 
in  LIGHT  that  ether-waves  can  be  polarized  so  that  all  the 
transverse  vibrations    take   place   in    one    direction ;   and 
such  waves  are  said  to  be  plane  polarized.     If  such  a  train 
of  ether-waves  is  sent  through  a  magnetic  field,  in  the 
direction  of  the  lines  of  force,  it  is  observed  that  the  direc- 
tion  in  which   the  vibrations   are    made    slowly  rotates. 
This  rotation  is  connected  with  the  direction  of  the  lines 
of  force  by  the  right-handed  screw  law;  and  it  is  niost 
pronounced  in  very  strong  fields  and  in  heavy  pieces  of 
glass. 

296,  The  "  Hall  Effect."    If  a  current  is  passed  through  a 
thin  sheet  of  metal,  such  as  foil  of  some  kind,  the  current 
spreads  out  and  flows  across  in  certain  definite  directions. 
It  is  always  possible  to  find  on  one  edge  of  the  thin  piece 
of  metal  a  point  which  has  the  same  potential  as  a  defi- 
nite point  on  the  other  edge ;  and,  in  fact,  lines  can  be 
drawn  across  the  sheet  such  that  all  the  points  on  any  one 


296]     GENERAL  PROPERTIES  OF  A  MAGNETIC  FIELD      387 

line  have  the  same  potential.  This  can,  of  course,  be  done 
by  means  of  a  galvanoscope,  which  would  detect  any  dif- 
ference of  potential. 
Suppose  that  two  such 
points  of  equal  poten- 
tial on  opposite  edges 
of  the  sheet  are  joined 
to  a  galvanoscope  ", 
there  will,  of  course, 
be  no  current  through 
it.  But  it  has  been 
found  by  experiment 
that  if  this  thin  sheet 
carrying  a  current  is 
placed  in  a  strong 
magnetic  field  perpen- 
dicular to  the  plane  of 
the  sheet,  e.  g.  between 
the  poles  of  an  elec- 
tro-magnet, a  current 
will  flow  through  the 
galvanoscope.  This 
proves  that  the  two 
points  on  the  sheet, 
which  had  the  same 
potential  before,  no  longer  have.  In  other  words,  as  a 
result  of  being  placed  perpendicular  to  a  magnetic  field, 
the  equipotential  lines  across  the  sheet  are  rotated  around 
an  axis  parallel  to  the  field,  so  as  to  lower  the  potential  of 
the  points  on  one  edge  of  the  sheet  and  raise  those  on  the 
other. 


FIG.  217. 


BOOK  V 
LIGHT 


.:  BOOK  v 

LIGHT 

INTRODUCTION 

297.  WHEN  the  nerves  of  the  retina  of  the  eye  are 
stimulated,  the  sensation  is  commonly  described  by  saying 
that  "  light "  is  seen.  Light  is,  then,  a  pure  sensation  ; 
and  it  may  be  caused  in  several  ways.  Ordinarily,  it  is 
produced  as  the  result  of  some  action  which  is  taking  place 
some  distance  away,  —  such  as  the  combustion  of  gas  in  a 
lamp-flame,  or  the  chemical  processes  on  the  sun. 

Various  colors  are  distinguished,  depending  upon  the 
nature  of  the  light-sensation ;  and  variations  in  the  inten- 
sity of  the  sensation  are  also  noticed. 

These  differences  are  not  confined  to  the  sensations  in 
the  eye.  Those  causes  which  produce  different  color  sen- 
sations also  produce  different  physical  effects  in  many 
cases,  and  have  different  properties.  Further,  if  two 
sources  of  light,  e.  g.  a  candle  and  a  gas  flame,  produce 
sensations  of  different  intensities  in  the  eye,  they  may  be 
proved  to  have  different  physical  properties.  The  source 
which  causes  the  stronger  sensation  will  produce  a  greater 
brightness  on  any  screen  which  it  illuminates,  or,  if  it 
produces  "  diffused  light "  through  any  semi-transparent 
substance  like  paraffin,  it  will  cause  a  greater  illumination 
than  the  weaker  source.  In  order,  then,  that  two  sources 
of  different  strengths  may  produce  equal  illuminations  in 


392  THEORY  OF  PHYSICS 

a  block  of  paraffin,  the  stronger  one  must  be  placed  farther 
off  than  the  weaker.  (This  statement  must  be  limited 
somewhat  in  certain  respects,  as  will  appear  later.) 

A  substance  is  said  to  be  "  opaque  "  if  it  allows  no  effect 
to  pass  through  it ;  "  transparent,"  if  it  does.  Applied  to 
"light,"  an  opaque  body  shuts  off  all  illumination;  or,  if 
opaque  to  only  certain  colors,  it  prevents  the  effect  pag- 
ing which  can  produce  those  sensations.  Similarly  a 
transparent  body  allows  all  or  only  certain  light-effects 
to  pass. 

It  is  also  known  to  all  that  sharp  light-shadows  are 
cast,  if  an  opaque  obstacle  of  any  kind  is  interposed  be- 
tween a  small  source  of  light,  such  as  a  candle,  and  a 
screen.  This  is  ordinarily  expressed  by  saying  that  "  light 
travels  in  straight  lines."  Again,  light,  considered  as  an 
effect  sent  out  by  some  source,  is  reflected  by  mirrors  and 
by  bodies  of  all  kinds.  When  light  falls  upon  a  rough 
body,  each  point  of  it  becomes,  as  it  were,  a  new  source  of 
light ;  and  it  is  owing  to  this  fact  that  most  natural  ob- 
jects are  seen.  Almost  every  one  is  probably  familiar  with 
the  fact  that,  when  light,  still  considered  as  an  effect  sent 
out  by  a  source,  falls  obliquely  upon  any  transparent  body, 
the  direction  of  its  propagation  in  this  body  is  changed. 
Illustrations  are  afforded  when  one  looks  into  a  basin  of 
water  obliquely  at  a  body  lying  on  the  bottom  ;  also  when 
a  lens  or  prism  is  interposed  between  a  source  of  light  and 
a  screen. 

All  these,  and  many  other  phenomena  associated  with 
light,  must  be  explained,  that  is,  must  be  shown  to  be  con- 
sequences of  a  simple  theory. 

298.  Ether- Waves.  It  is  now  known,  as  the  result  of 
experiments  which  will  be  described  in  Chapter  I.,  that 
the  sensation  light  is  produced  when  certain  waves  enter  the 
eye,  these  waves  having  been  produced  in  the  ether  by  the 
actions  in  the  so-called  source  of  light.  It  will  be  proved 
that  these  waves  are  transverse,  and  that  waves  of  differ- 


LIGHT  393 

ent  periods  produce  in  the  eye  sensations  of  different  colors. 
That  these  waves  are  in  the  ether  may  be  considered  as 
demonstrated  by  the  fact  that,  so  far  as  our  experiments 
enable  us  to  test  the  point,  the  presence  of  ordinary  mat- 
ter is  not  in  the  least  essential  for  the  propagation  of  the 
waves.  Further,  the  waves  advance  with  a  finite  velocity 
which  can  be  measured  without  difficulty,  as  will  be  shown 
in  Chapter  I.,  which  proves  the  existence  of  a  medium  en- 
dowed with  inertia.  Of  course,  as  might  be  expected,  the' 
presence  8f  ordinary  matter  in  the  ether  loads  it,  and  so 
influences  the  velocity  of  propagation  of  the  waves,  —  al- 
ways making  them  go  more  slowly.  Waves  in  the  ether 
may  have  different  lengths  depending  upon  the  frequency 
of  the  vibration  which  produces  them  ;  but,  as  will  be 
proved  later,  the  velocity  of  all  waves  in  the  pure  ether  is 
the  same  ;  while,  when  there  is  matter  immersed  in  the 
ether,  the  longer  waves  will  be  less  affected  than  the 
shorter  ones,  and  so  will  have  a  greater  velocity.  (There 
are  certain  exceptions  in  strongly  absorbing  media.) 

Of  these  ether-waves  only  those  which  have  lengths  be- 
tween rather  narrow  limits  can  produce  the  sensation  light ; 
but  there  are  other  waves  both  longer  and  shorter.  Those 
which  can  produce  light  may,  of  course,  be  more  easily 
observed  and  studied  than  the  others,  because  every  nor- 
mal man  is  provided  by  nature  with  an  instrument,  the 
eye,  which  enables  him  accurately  to  observe  and  measure 
many  of  the  phenomena  of  light.  Those  laws  of  reflection, 
refraction,  dispersion,  interference,  polarization,  etc.,  which 
are  deduced  by  observations  on  light-waves  are  also  true 
for  longer  and  shorter  waves  ;  the  only  reason  why  they  are 
demonstrated  for  light-waves  in  preference  to  the  others 
being  the  great  ease  with  which  these  may  be  observed. 

Ether-waves  which  are  too  long  to  produce  the  sensa- 
tion light  may  be  studied  by  various  methods  described  in 
HEAT,  depending  upon  the  fact  that  all  ether-waves  are 
carrying  energy.  Again,  those  waves  which  are  too  short 

13* 


394  THEORY  OF  PHYSICS 

to  produce  the  sensation  light  may  be  studied  by  photo- 
graphic methods,  because  all  ether-waves,  and  especially 
the  short  ones,  can  produce  chemical  changes  in  certain 
compounds. 

299.  Properties  of  Waves.  It  may  be  well  to  give  a  brief 
statement  of  the  properties  of  waves  in  general ;  and  the 
student  is  advised  to  read  the  chapters  in  MECHANICS, 
SOUND,  and  HEAT,  which  explain  various  kinds  of  waves. 
Waves  are  produced  by  the  vibrations  of  the  source ;  in 
ether-waves  the  matter  may  be  in  vibration,  and  by  its 
action  on  the  ether  send  out  waves  in  it.  If  the  source  is 
a  point,  the  wave-front  will  be  a  sphere ;  and  the  intensity 
of  the  waves  will  vary  inversely  as  the  square  of  the 
distance  from  the  source.  (This  last  fact  permits  the 
strengths  of  two  sources  of  light  to  be  compared  and 
measured,  as  already  explained,  by  means  of  a  so-called 
paraffin  "  photometer.")  If  the  wave-front  is  a  sphere,  the 
waves  are  called  "  spherical ;  "  and,  if  it  is  a  plane,  they  are 
called  "  plane." 

The  number  of  vibrations  per  second  of  the  source  is 
called  the  "  frequency ; "  and  it  equals  the  number  of 
"  wave-crests "  sent  out  in  one  second,  or  the  "  wave- 
number,"  n.  The  distance  between  one  "  crest "  and  the 
next  is  called  the  "  wave-length,"  X  ;  and  the  distance  the 
waves  advance  in  one  second  is  called  the  velocity  of 
the  wave,  v ;  so  that  v  —  n  X.  In  a  pure  medium,  e.  g.  the 
pure  ether,  the  velocity  is  independent  of  everything  ex- 
cept the  elasticity  and  inertia. 

The  primary  wave  may  also  be  considered  as  replaced  at 
any  instant  by  secondary  waves,  as  explained  in  Article  75. 
This  is  true  of  a  spherical  wave  or  of  a  plane  wave. 


CHAPTEE   I 


THE  WAVE-THEORY 

BY  the  wave-theory  is  meant  the  statement  that  the 
phenomena  of  light  (and  all  those  due  to  other  radiations 
in  the  ether*  heat,  chemical,  electrical,  etc.)  are  produced 
by  waves  in  the  ether.  These  waves  will  be  proved  later, 
in  Chapter  VIII.,  to  be  transverse  ;  but  for  all  present 
purposes  it  is  entirely  immaterial  whether  the  waves  are 
transverse  or  longitudinal.  The  proof  of  the  existence  of 
these  waves  and  of  the  intimate  connection  between  them 
and  the  sensation  light  is  furnished  by  any  so-called 
"  interference  "  experiment. 

300.  Young's  Interference  Experiment.  Three  parallel 
opaque  screens  are  placed  some  few  inches  apart;  in  the 
first  is  made  a  narrow  slit  or 
opening  with  straight  parallel 
edges,  S;  in  the  second,  there 
are  made  two.  narrow  slits  par- 
allel to  that  in  the  first  screen 
and  quite  close  together,  A  and  B. 
There  is  thus  a  narrow  opaque 
portion  between  the  two  slits  A 
and  B ;  and  the  two  screens  are 
so  adjusted  that  when  a  source 
of  light,  such  as  a  candle,  is  placed  in  front  of  the  first 
slit,  the  candle,  the  first  slit,  and  the  opaque  portion  be- 
tween the  two  slits  are  in  the  same  straight  line.  If  this 
is  done,  the  two  slits  A  and  B  are  equally  illuminated  by 
light  from  the  slit  S,  but  there  will  not  be  uniform  illu- 


FIG.  218. 


396  THEORY   OF  PHYSICS  [CH.  I 

initiation  over  the  third  screen  which  is  receiving  light 
from  the  two  slits  A  and  B.  It  may  be  observed  that  on 
this  third  screen  there  are  "  bands,"  parallel  to  the  slits  ; 
that  is,  strips  of  the  screen  are  illuminated,  but  in  between 
these  strips  there  is  darkness.  These  bands  will  be  at 
fairly  equal  distances  apart,  if  the  source  of  light  is  an 
incandescent  electric  lamp ;  and,  if  a  piece  of  colored  glass 
is  interposed  between  the  source  and  the  slit  S,  the  bands 
are  at  exactly  equal  intervals  for  quite  a  wide  range.  (In- 
stead of  having  the  third  screen,  an  equally  good  method 
is  simply  to  receive  the  light  from  the  two  slits  A  and  B 
in  the  eye  directly.)  In  any  case,  it  may  be  necessary  to 
use  a  microscope  in  order  to  see  the  bands  clearly. 

The  only  satisfactory  explanation  of  this  experiment  is 
as  follows :  the  ether- waves  from  the  slit  S  reach  the  t\vo 
slits  A  and  B,  thus  making  them  two  sources  of  waves, 
which  are  identical  in  all  respects,  if  proper  precautions 
are  observed.  The  two  sets  of  waves  which  proceed  out 
from  the  slits  A  and  B  illuminate  the  third  screen ;  but, 
to  reach  the  same  points  on  the  screen,  the  two  trains  of 
waves  must  go  different  distances,  in  general.  If  there  is 
a  point  on  the  screen  whose  distance  from  A  is  r,  and 

whose  distance  from  B  is  r  ±  -  ,  where  \  is  the  wave- 
length of  the  waves  emitted  by  the  source,  there  will  be 
no  effect  at  all  at  that  point,  because  one  train  of  waves 
will  completely  neutralize  the  effect  of  the  other :  the 
disturbances  in  a  wave  at  a  distance  apart  of  half  a  wave- 
length are  exactly  opposite  each  other.  The  central  por- 
tion of  the  third  screen,  that  is,  the  portion  in  line  with 
the  slit  S,  and  the  opaque  strip  between  A  and  B  will  be 
illuminated,  because  the  two  trains  of  waves  reinforce 
each  other ;  but  on  each  side  of  this  bright  portion  there 
will  be  a  series  of  points  such  as  described  above,  where 

the  difference  in  path  of  the  two  trains  of  waves  is     ,  and 


300]  THE   WAVE-THEORY  397 

there  is  darkness.  There  is  thus  a  bright  central  band 
parallel  to  the  slits,  and  a  dark  band  on  each  side.  On 
the  farther  side  of  each  of  these  dark  bands  there  will  be 
a  bright  band  made  up  of  points  such  that  their  distances 
from  the  two  slits  differ  by  a  whole  wave-length  X ;  for 
under  these  conditions  the  waves  strengthen  each  other. 
Again,  on  the  farther  side  of  each  of  these  bright  bands, 
there  is  a  dark  band,  for  all  the  points  of  which  the  dis- 

3  X 

tances  to  the  two  slits  differ  by  — ,  etc.     So  it  is  at  once 

& 

evident  that  by  the  wave-theory  it  is  possible  to  com- 
pletely explain  the  interference-bands ;  and  on  no  other 
theory  can  it  be  done. 

It  is  evident,  also,  that  the  distance  apart  of  the  bands 
must  depend  upon  the  length  of  the  waves ;  for,  the 
greater  X  is,  so  much  the  farther  must  the  bright  bands 
be  separated,  in  order  that  the  distances  to  the  two  slits  may 
differ  by  X,  2  X,  etc.  It  may  be  proved  directly  by  ex- 
periment that,  when  red  light  illuminates  the  slits,  the 
bands  are  farther  apart  than  when  blue  light  is  used, 
thus  proving  that  red  light  has  a  wave-length  greater  than 
that  of  blue  light.  If  all  the  common  colors  are  examined, 
it  can  be  proved  that  they  may  be  arranged  in  the  follow- 
ing order  of  increasing  wave-length :  violet,  blue,  green, 
yellow,  orange,  red.  If  ordinary  white  light  is  used,  the 
bands  are  seen  to  be  overlapping  colored  bands ;  and,  in 
fact,  the  experiment  proves  that  white  light  is  a  mixture 
of  waves  of  all  possible  wave-lengths  (or  wave-numbers), 
which  are  separated  by  the  interference  into  their  corre- 
sponding bands.  Any  source  of  light  which  emits  a  train 
of  waves  of  a  constant  wave-number  is  said  to  produce 
"  homogeneous  "  waves. 

(It  is  obvious  that  interference-bands    are   not  a  phe- 
nomenon of  light,  but  of   waves.     Sound-waves  can  give 
interference-bands;   and   so    can   ether- waves,   no   matter' 
what  their  length  is:  the  production  and  appearance  of 


398  THEORY  OF  PHYSICS  [CH.  I 

the  bands  is  purely  a  question  of  the  dimensions  of  the 
apparatus.) 

301.  Velocity  of  Light.  These  waves  which,  if  of  suit- 
able wave-number,  can  produce  the  sensation  light,  are 
phenomena  of  the  ether,  as  is  proved  by  many  facts. 
Various  stars  emit  these  waves  which  cause  light  in  our 
eyes ;  and  there  is  good  evidence  for  believing  that  no 
ordinary  matter  exists  in  the  space  between  the  stars  and 
the  earth;  therefore,  some  medium  which  is  capable  of 
carrying  waves  must  fill  this  space,  and  this  has  been 
called  the  ether.  That  the  ether  has  inertia  is  proved  by 
the  fact  that  the  waves  in  it  travel  with  a  finite  velocity. 
This  velocity  is  very  great,  being  3  X  1010  cm.  per  second, 
or  about  180,000  miles  per  second;  but  its  value  may  be 
quite  accurately  measured.  The  first  observation  which 
led  to  the  determination  of  the  so-called  "velocity  of 
light "  was  the  fact  that  certain  discrepancies  were  noticed 
between  the  calculated  and  the  observed  times  of  eclipse 
of  Jupiter's  satellites.  The  planet  Jupiter  has  several 
satellites,  which  make  regular  revolutions  around  it,  just 
as  our  moon  does  around  the  earth.  It  seems  perfectly 
easy,  then,  to  calculate  when  the  satellite  will  disappear 
behind  Jupiter,  i.  e.  be  eclipsed ;  but  it  was  noted  that  the 
times  of  eclipse  calculated  several  months  in  advance  did 
not  agree  with  the  observations.  The  explanation  was 
most  simple :  at  the  end  of  several  months  the  earth  has 
moved  until  it  is  on  the  opposite  side  of  its  orbit  from 
where  it  was  before;  and  in  one  of  these  positions  the 
waves  must  travel  across  the  diameter  of  the  earth's  orbit 
in  order  to  reach  it.  This  requires  some  minutes ;  and  the 
difference  between  the  predicted  and  the  observed  times  of 
eclipse  may  be  accurately  determined.  If  the  diameter  of 
the  orbit  is  known,  and  the  number  of  seconds  required 
for  the  waves  to  pass  across,  the  velocity  of  light  may  be 
at  once  calculated. 

Another  method  for  the  determination  of  this  velocity 


301] 


THE   WAVE-THEORY 


399 


was  devised  by  Fizeau,  and  is  known  as  his  method. 
Light-waves  from  a  slit  S  are  allowed  to  fall  upon  a  glass 
mirror,  M,  which  reflects  them  at  right  angles  so  that  they 
just  graze  the  edge  of  a  large-toothed  wheel,  W.  If  the 
wheel  is  so  placed  that  a  tooth  intercepts  the  light,  none 
passes ;  but,  if  an  opening  is  in  the  path,  the  light- waves 
pass  through,  and  by  means  of  lenses  are  directed  to  a  dis- 


FIG.  219. 

tant  concave  mirror,  C,  which  reflects  them  back  over  the 
same  path.  If  the  wheel  is  turned  rapidly,  it  may  happen 
that  the  waves  on  their  return  reach  the  wheel  just  as  a 
tooth  blocks  the  way ;  while,  if  the  rotation  is  still  more 
rapid,  the  next  opening  in  the  wheel  may  be  in  the  path, 
and  the  return- waves  can  pass  through.  They  will  then 
fall  upon  the  glass  mirror,  M ;  and  some  light  will  pass 
through  in  a  straight  line,  and  may  be  observed  at  0.  If 
the  width  of  a  tooth  and  an  opening,  the  number  of  revo- 
lutions of  the  wheel  per  second,  and  the  distance  between 
the  wheel  and  the  concave  reflecting  mirror  are  known, 
the  velocity  of  the  waves  may  be  at  once  calculated. 

Another  and  more  accurate  method  is  one  devised  by 
Foucault,  and  called  by  his  name.  Waves  from  a  source 
of  light  at  a  slit  S  pass  through  a  glass  mirror  M,  fall 
upon  a  mirror  R,  and  are  reflected  to  a  distant  concave 
mirror  C,  which  reflects  them  back  over  their  same  path  ; 
but,  just  before  they  reach  the  slit,  they  fall  upon  the 


400 


THEORY  OF  PHYSICS 


[CH.  I 


glass  mirror  M,  which  reflects  some  of  them  one  side  to  a 
point,  0,  where  they  can  be  observed.  But,  if  the  mirror 
R  is  revolving  about  a  vertical  axis,  it  will  have  changed 
its  position  slightly  during  the  time  the  waves  took  for 
their  passage  from  R  to  C,  and  back  again ;  and  conse- 


FIG.  220. 


quently  the  return-waves  will  not  be  brought  to  the  same 
point,  0,  to  which  they  would  have  come  if  the  mirror  R 
had  been  at  rest.  If  the  speed  of  rotation  of  the  revolving 
mirror  R,  the  distance  from  R  to  C,  and  the  shift  of  the 
point  of  light,  0,  are  known,  the  velocity  of  light  may  be 
calculated. 

The  mean  of  the  best  results  is,  as  said  above,  3  X  1010 
cm.  per  sec.  This  is  the  velocity  of  ether-waves  of  all 
wave-lengths,  because,  as  noted  several  times  (see  Arts.  74 
and  119),  the  velocity  of  waves  of  any  definite  kind  in  a  pure 
medium  is  independent  of  the  wave-length.  For  light, 
this  fact  is  demonstrated  by  the  observation  that,  when  a 
white  star  is  eclipsed  by  another,  it  appears  white  immedi- 
ately before  and  after  the  eclipse.  If  red  light  had  a 
greater  velocity  than  blue,  the  star  would  have  appeared 
blue  just  before  it  disappeared,  and  red  as  soon  as  it 
reappeared  ;  but  neither  this  nor  the  converse  is  the  case. 

When  the  waves  enter  ether  in  which  there  is  ordinary 
matter,  the  velocity  is  naturally  changed;  and  Foucault 
proved  by  direct  experiment  that,  if  water  is  the  matter, 
the  velocity  is  lessened.  This  is  now  known  to  be  true 


302] 


THE   WAVE-THEORY 


401 


for  all  matter,  including  air ;  and  so  a  correction  must  be 
applied  to  Fizeau's  and  Foucault's  results  in  order  to 
determine  the  velocity  in  the  pure  ether.  The  correction 
is,  however,  very  small  when  air  is  the  matter  immersed 
in  the  ether,  being  about  three  parts  in  ten  thousand.  As 
might  be  expected,  waves  of  different  wave-lengths  do  not 
travel  with  the  same  velocity  in  ether  thus  loaded  with 
matter.  The  general  law  is,  that  the  less  the  wave-num- 
ber, so  much  the  less  is  the  velocity  affected.  Thus  red 
light  is  not  influenced  so  much  as  is  blue.  In  the  case  of 
air,  the  difference  for  red  and  blue  light  is  so  slight  that  it 
cannot  be  noticed  in  any  ordinary  experiments. 

302.  Rectilinear  Propagation.  When  plane  waves  are 
advancing  in  any  direction,  they  may  be  considered  re- 
placed at  any  instant, 
as  already  said,  by 
secondary  spherical 
waves  emitted  by  each 
point  of  the  wave- 
front  ;  and  the  surface 
which  is  tangent  to 
these  secondary  waves 
will  be  the  advancing 
wave-front.  The  dis- 
turbance at  any  point, 
P,  in  the  path  of  the 
wave  will  then  be  due 
to  spherical  waves  sent  out  by  all  the  points  of  the  pri- 
mary wave-front  which  was  resolved  into  secondary  waves. 
It  is  of  importance  to  learn  whether  all  of  these  secondary 
waves  influence  the  disturbance  at  P,  or  whether  some  of 
them  neutralize  each  other.  Draw  from  P  a  perpendicular 
line  upon  the  primary  wave-front,  and  let  it  intersect  the 
plane  at  0,  which  is  -called  the  pole  of  P.  Call  the  dis- 
tance P  0,  r.  With  radii 


FIG.  221. 


402 


THEORY  OF  PHYSICS 


[CH.  I 


r+  -z, 


+  X,    r 


1*. 


2X,   etc., 


where  X  is  the  wave-length  of  the  waves,  describe  spheres 

around  P  as  a  centre.  They 
will  intersect  the  plane  wave- 
front  in  circles,  as  shown,  all 
concentric  around  the  pole  0. 
The  plane  will  thus  be  divided 
into  zones  included  between 
circles  ;  and  these  are  called 
"  Huyghens'  zones."  It  is  evi- 
dent that  a  spherical  wave 
emitted  from  a  point  in  any 
zone  will  have  a  distance  to 
pass,  in  order  to  reach  P,  which 

is  exactly  --   shorter  than  the 


FIG.   222. 


distance  for  a  definite  point  in  the  zone  just  outside  it, 
and  is  ^  longer  than  the  distance  for  a  definite  point  in 

the  zone  just  inside  it.  It  follows  from  this  that  the  effect 
at  P  due  to  the  secondary  waves  from  any  one  zone  is  of 
the  opposite  kind  from  the  effects  due  to  the  waves  coming 
from  the  two  contiguous  zones  ;  for,  since  the  paths  differ 

by  _,  the  displacements  at  P  will  be  in  opposite  direc- 

A 

tions.  The  total  effect  at  P  due  to  all  the  zones,  i.  e.  to 
the  entire  series  of  secondary  waves,  may,  then,  be  written, 

/•=  m\  —  mz  +  ms  —  m*  +  ^5  —  wze  +  etc.  .     .    (1) 

where  mi  is  the  effect  due  to  the  first  zone,  the  central 
one  ;  w2,  that  due  to  the  second,  etc.  The  numerical  value 
of  m,  the  effect  due  to  any  zone,  depends  upon  three 
things  :  the  area  of  the  zone,  the  distance  of  the  zone  from 
P,  the  inclination  of  the  waves  to  the  line  OP.  It  may 
be  proved  that  these  first  two  conditions  so  neutralize 


303]  THE   WAVE-THEORY  403 

each  other  that  the  effect  at  P  depends  only  upon  the 
third.  The  more  inclined  to  the  line  OP  the  waves  are, 
so  much  the  less  effect  do  they  produce ;  therefore,  the 
numerical  value  of  any  m  is  greater  than  that  of  the  suc- 
ceeding one.  The  quantity  /  =  mi  —  ra2  +  ms  —  m^  +,  etc. 
is,  then,  a  series  of  decreasing  terms,  with  alternating  signs, 
the  differences  between  consecutive  terms  being  extremely 
small.  The  series  may  be  arranged  as  follows :  — 

/  =  i  mi  +  1  (I7ll  —  w2)  —  1  (m2  —  ms)  +  1  (ms  —  ra4)  — ,  etc. 
But  mi  —  ?;i2,  w2  —  wg,  etc.  are  so  small  that 

J  (ml  —  m2)  —  1  (mz  —  m8)  +  £  (ra8  —  w4)  — ,  etc. 
is  practically  zero  in  comparison  with  ^  mi.     Consequently, 
/=  Jwi (2) 

Expressed  physically  this  means  that  the  effect  of  any  one 
zone  is  completely  neutralized  by  the  action  of  half  of 
each  of  the  zones  on  the  two  sides  ;  so  that  the  entire 
effect  at  P  is  due  to  the  secondary  waves  emitted  by  the 
first  central  zone  around  the  pole  0.  The  wave-length  of 
ordinary  light-waves  is  about  ^5^6()th  of  an  inch;  and  the 
area  of  the  first  zone  must,  therefore,  be  so  small  as  al- 
most to  be  regarded  as  a  geometrical  point. 

Corresponding,  therefore,  to  any  point  in  the  wave- 
front,  there  is  a  series  of  points  in  the  path  of  the  waves, 
the  disturbances  at  which  will  be  due  entirely  to  the 
secondary  waves  coming  from  the  first  point.  If  the 
waves  do  not  pass  into  a  different  medium,  these  points 
will  all  lie  in  a  straight  line,  called  a  "  ray."  This  phe- 
nomenon is  sometimes  described  as  the  "  rectilinear  propa- 
gation of  light."  A  line  drawn  perpendicular  to  the  wave- 
front  is  called  the  "  wave-normal ; "  and,  if  the  medium  in 
which  the  waves  are  advancing  is  isotropic,  the  ray  and 
the  wave-normal  coincide. 

303.  Shadows.  It  is  easy,  then,  to  explain  why  sharp 
shadows  are  cast  by  opaque  obstacles,  if  there  is  but  one 


404 


THEORY  OF  PHYSICS 


[CH.  I 


source  of  light.  If  a  plane  wave  is  advancing  from  A  to- 
wards B,  and  meets  there  an  opaque  obstacle,  each  point 
of  the  wave-front  0,  0',  0" ',  etc.,  will  produce  effects  at  the 
points  P,  P',  P",  etc.  on  a  screen,  O,  if  0  P,  0'  P',  etc., 
are  lines  perpendicular  to  the  wave-front.  If  0  is  the 
point  on  the  wave-front  just  above  the  obstacle,  its  corre- 
sponding point,  P,  will  be  the  lowest  point  on  the  screen 


"  \* 

FIG.  223. 

to  receive  waves ;  and  so  there  should  be  a  sharp  shadow 
below  this  point.  It  must  be  noticed,  though,  that  if 
Huyghens'  zones  are  drawn  around  0,  as  the  pole  of  P, 
portions  of  these  zones  will  be  below  the  edge  of  the 
obstacle,  and  so  cannot  produce  any  neutralizing  effect  at 
P.  It  would  be  expected  from  this  that  the  illumination 
and  general  phenomena  at  the  edge  of  the  shadow  would 
not  be  so  simple  as  at  points  inside  and  well  outside  the 
shadow.  Such  is  actually  the  case,  as  will  be  explained 
later.  The  phenomena  at  the  edge  of  shadows  are  in- 
cluded in  the  name  "diffraction"  phenomena;  but,  for 
general  purposes,  they  may  be  neglected,  and  opaque  ob- 
stacles may  be  considered  as  casting  sharp  shadows.  If 
there  are  two  sources  of  light,  the  phenomena  are  slightly 
more  complicated,  because  in  this  case  the  shadow  cast  by 
an  opaque  obstacle  placed  in  the  field  of  light  of  one 
source,  Si,  is  illuminated  in  certain  regions  by  the  light 


304] 


THE   WAVE-THEORY 


405 


FIG.  224 


from  the  other  source,  $>,  and  vice  versa.  Other  portions 
of  space  will  receive  no  light  at  all,  while  still  others  will 
be  illuminated  by  both 
sources.  This  is  shown 
in  the  diagram,  where 
the  black  portion,  which 
receives  no  waves  at  all, 
is  called  the  "  umbra ;  " 
the  shaded  portions, 
each  of  which  receives 
waves  from  only  one 
source,  the  •'  penumbra." 
A  still  more  compli- 
cated case  is  when  the 
source  of  light  is  not  a 

point,  but  of  considerable  size,  like  a  candle-flame  or 
the  sun.  If  an  obstacle  is  interposed  in  the  path  of  the 
waves,  there  will  be,  however,  as  before,  umbra  and  pe- 
numbra. A  very  special  illustration  is  given  when  the 
moon  passes  between  the  sun  and  the  earth ;  if  the  earth 
is  in  the  umbra,  the  eclipse  is  said  to  be  "  total ;  "  if  in  the 
penumbra,  "  partial." 

304.  "Pin-hole  Im- 
ages." Another  illus- 
tration of  the  recti- 
linear propagation  of 
light  is  afforded  by 
what  are  known  as 
"  pin-hole  images." 
A  small  opening,  e.  g. 
a  pin-hole,  is  made  in 
an  opaque  screen  ; 
and  any  illuminated 
object,  such  as  an 

arrow  on  which  sunlight  is  falling,  is  placed  in  front  of 
the  opening.  Each  point,  A,  of  the  arrow  sends  out 


FIG.  225. 


406  THEORY  OF  PHYSICS  [CH.  I 

waves ;  but  only  a  small  cone  of  them  passes  through 
the  pin-hole.  A  corresponding  bright  spot  is  thus  formed 
on  a  screen  at  B,  the  spot  having  the  same  shape  as  the 
pin-hole  opening.  If  the  arrow  is  quite  far  from  the 
screen  which  has  the  opening,  and  if  the  screen  which 
receives  the  light  is  quite  near  it,  the  illuminated  spot  will 
be  small.  Each  point  of  the  arrow  produces  a  correspond- 
ing bright  spot  on  the  screen  ;  and  they  will  all  slightly 
overlap,  so  that  the  shape  of  the  pin-hole  opening  is  im- 
material. There  will,  therefore,  be  on  the  receiving  screen 
a  fairly  sharp,  inverted  image  of  the  arrow.  This  is  the 
principle  of  pin-hole  photography,  where  the  pin-hole 
takes  the  place  of  a  lens ;  and  it  also  explains  the  appear- 
ance of  the  images  of  the  sun  which  are  seen  under  trees 
whose  leaves  are  thick  and  close,  because  in  this  case  the 
"  pin-hole  "  is  some  small  opening  between  the  leaves. 


CHAPTEE   TI 


REFLECTION 

As  explained  in  SOUND  (Art.  135),  whenever  waves  in 
one  medium  reach  a  surface  separating  it  from  another, 
there  will,  in  general,  be  reflected  waves ;  and  certain  im- 
portant differences  were  noted  between  reflection  from  a 
medium  in  which  the  velocity  of  propagation  is  greater 
than  in  the  original  medium,  and  the  similar  case  when 
the  velocity  is  less. 

305.  Mirrors.     A  mirror  is  a  smooth  surface  separating 
two  media  in  which  the  velocity  of  the  waves  is  different. 
"Smoothness"  is  a  purely  relative  term;  but  here  it  is 
meant  to  imply  that  there  are  no  unevennesses  which  are 
at  all  comparable  in  size  with  the  wave-length  of  the  waves. 
The  commonest  forms  of  mirrors  are  those  whose  surfaces 
are  either  plane  or  spherical ;  and  they  are  called,  accord- 
ingly, plane  or  spherical  mirrors. 

There  are  several  cases  of  reflection  to  be  considered, 
depending  upon  the  form  of  the  wave-front  and  that  of 
the  mirror. 

306.  Plane  Waves  Incident  on  a  Plane  Mirror.    Let  the  in- 
cident plane  waves  meet 

the  plane  mirror  in  a 
line  perpendicular  to  the 
plane  of  the  paper ;  then 
the  section  of  the  plane 
wave-front  maybe  00'0"t 
and  that  of  the  mirror 
0  Q  P' ;  and  their  plane 
is  called  the  "  plane  of  incidence."  When  the  wave-front 


408  THEORY   OF   PHYSICS  [CH.  II 

has  the  position  0  0'  0",  it  may  be  considered  as  replaced 
by  secondary  waves  ;  and  the  wave-front  at  any  future  time 
will  be  the  surface  which  is  tangent  to  all  these  secondary 
waves.  The  disturbance  from  0"  proceeds  in  a  straight  line 
perpendicular  to  the  wave-front,  as  explained  in  the  last 
chapter  (Art.  302),  and  reaches  a  point,  P" t  on  the  mirror 
after  a  certain  time.  Let  the  form  and  position  of  the  wave- 
front  at  the  end  of  this  time  be  determined.  It  must  evi- 
dently pass  through  P"  and  a  line  drawn  perpendicular  to 
the  plane  of  the  paper  at  that  point.  Draw  from  0  a  line  OP, 
making  the  angle  P  0  P"  equal  to  the  angle  0"  P"  0,  and 
from  P"  draw  a  line  perpendicular  to  0  P;  then  this  line  may 
be  proved  to  be  a  section  of  the  reflected  wave-front  by  the 
plane  of  the  paper.  The  triangles  0"  P'  0  and  P  0  P" 
are  equal ;  therefore  the  lines  0  P  and  0"  P"  are  equal. 
But  while  the  disturbance  has  gone  from  0"  to  Pn,  the  sec- 
ondary spherical  waves  from  0  have  spread  out  until  the 
wave-front  has  an  equal  radius  0"  P"  (or  0  P).  There- 
fore, since  P"P  is  perpendicular  to  0  P,  it  is  tangent  to 
the  sphere  drawn  around  0  as  a  centre,  and  with  a  radius 
0  P.  Similarly,  the  disturbance  from  another  point,  0', 
reaches  the  mirror  at  Q  when  the  disturbance  from  0", 
reaches  Q',  where  Q  Qf  is  a  line  parallel  to  the  section  of 
the  wave-front  0  0' 0"  \  and,  while  the  disturbance  pro- 
ceeds from  Q'  to  P" ',  that  from  Q  spreads  out  in  the  form 
of  a  sphere  with  a  radius  equal  to  Q'  P".  If  a  line  QPf  is 
drawn  perpendicular  to  Pn  P,  it  is  evidently  equal  in 
length  to  Qf  P" ;  hence  the  plane  surface  whose  section  is 
P"  P1  P  will  be  tangent  to  the  secondary  wave  from  Q. 
Consequently  the  reflected  wave-front  is  a  plane  surface 
which  intersects  the  plane  mirror  in  a  line  perpendicular 
to  the  plane  of -the  paper;  and  its  section  with  the  paper 
is  the  line  P"  P'  P. 

The  "  angle  of  incidence  "  is  defined  as  the  angle  be- 
tween the  normal  to  the  incident  wave-front  and  that  to 
the  plane  mirror,  i.  e.  it  is  0"  OP".  Similarly  the  "angle 


307]  REFLECTION  409 

of  reflection "  is  the  angle  between  the  normal  to  the  re- 
flected wave-front  and  that  to  the  mirror,  i.  e.  P  P"  0. 
These  two  angles  are  evidently  equal.  Consequently,  from 
these  last  two  paragraphs  the  laws  of  ordinary  reflection 
follow  at  once  :  — 

1.  The  normals  to  the  incident  and  reflected  wave-fronts 
and  to  the  plane  mirror  all  lie  in  one  plane,  viz.  the  plane 
of  incidence. 

2.  The  angle  of  incidence  equals  the  angle  of  reflection. 
These  laws,  thus  deduced  on  the  wave-theory,  are  veri- 
fied by  direct  experiment. 

It  should  be  noted  that  each  point  in  the  incident  wave- 
front  corresponds  to  a  definite  point  in  the  reflected  wave- 
front  ;  so  that  the  effect  at  the  latter  point  is  entirely  due  to 
the  disturbance  at  the  former.  (This  can,  of  course,  be  proved 
in  detail  by  drawing  Huyghens'  zones,  as  was  done  in  the 
case  of  rectilinear  propagation.)  Thus  P  corresponds  to 

0,  P'  to  Of,  P"  to  0".    It  may  be  shown,  too,  that  the  path 
Or  Q  P'  is  the  shortest  broken  line  touching   the   mirror 
which  can  be  drawn  from  0'  to  P' . 

It  is  hardly  necessary  to  remark  that  this  proof  is  not  a 
demonstration  for  a  phenomenon  of  light,  but  applies  to 
any  train  of  waves  falling  upon  a  piano  mirror  of  suitable 
size.  It  is  true  of  sound-waves  and  all  others. 

307.  Rotating  Plane  Mirror.  If  plane  waves  fall  nor- 
mally upon  a  plane  mirror,  i.  e.  if  the  angle  of  incidence 
is  zero,  waves  will  be  reflected  directly  back  in  the  same 
direction.  If  the  mirror  is  now  turned  through  an  angle 
0  around  an  axis  perpendicular  to  the  plane  of  incidence, 

1.  e.  if  the  normal  to  the  mirror  now  makes  the  angle  0 
with  the  normal  to  the  incident  waves,  the  normal  to  the 
reflected  waves  must  also  make  the  same  angle,  0,  with 
the  normal  to  the  mirror,  but  on  the  opposite  side,  because 
the   angle   of   reflection    equals    the   angle    of    incidence. 
Therefore,  by  turning  the  mirror  through  an  angle  9,  the 
direction  of  the  reflected  waves  has  been  turned  through 


410 


THEORY  OF  PHYSICS 


[CH.  II 


FIG.  227. 

an  angle  2  6.  This  mathematical  principle  of  a  rotating 
mirror  is  made  use  of  in  many  instruments,  such  as  the 
sextant,  reflecting  mirror-galvanometers,  etc. 

308.  Spherical  Waves  Incident  on  a  Plane  Mirror.    If  there 
is  a  source  of  waves  at  a  point  0,  spherical  waves  will  be 

sent  out ;  and  if  there  is  a 
plane  surface  near  by,  the 
waves  will  reach  it  at  a  point 
M,  where  0  M  is  perpen- 
dicular to  the  surface.  If  the 
surface  were  not  there,  the 
waves  would  in  a  certain  time 
reach  the  position  B  A  C ; 
but,  owing  to  the  presence  of 
the  surface,  when  the  wave- 
front  reaches  M,  a  secondary  spherical  wave  is  sent  back, 
which,  in  the  time  taken  for  the  disturbance  to  reach  B 
and  C,  has  spread  to  a  radius  M  A',  which  equals  MA. 
The  reflected  wave-front  at  this  instant  must  include  the 
points  B  and  C,  and  must  be  tangent  to  the  sphere  of 
radius  MA1  drawn  around  M.  Such  a  surface,  if  spheri- 


308]  REFLECTION  411 

cal,  must  have  a  centre  0'  at  the  same  distance  perpen- 
dicularly below  the  surface  at  M  as  0  is  above  it.  It  may 
be  proved  without  difficulty  that  this  spherical  surface  is 
really  tangent  to  all  the  secondary  waves  emitted  by 
the.  various  points  on  the  mirror  around  M  as  the  dis- 
turbance reaches  them ;  and  so  it  is  actually  the  reflected 
wave-front. 

This  is  described  by  saying  that,  when  waves  are  emitted 
from  a  point  0  and  fall  upon  a  plane  surface,  they  seem  to 
come  after  reflection  from  a  point  (7,  where  O  0  is  a  line 
perpendicular  to  the  surface  and  bisected  by  it.  Such  a 
point  0'  is  called  a  "  virtual  image  "  of  0 ;  "  virtual,"  be- 
cause the  waves  after  reflection  do  not  actually  come 
through  or  from  (/,  but  only  seem  to. 

If  any  illuminated  object,  e.  g.  an 
arrow,  is  held  above  a  plane  mirror, 
each  point  of  it  will  have  an  image 
in  the  mirror ;  and  there  will  be  a 
virtual  image  of  the  arrow  from 
which  the  light  will  seem  to  come, 
if  the  waves  are  reflected  from  the 
mirror.  This  image  will  have  the 
same  dimensions  as  the  illuminated 
object ;  that  is,  there  is  no  magnifi-  FIG.  229. 

cation. 

Even  if  the  source  of  waves  is  to  one  side  of  the  plane 
mirror,  the  same  construction  and  theory  applies,  because 
this  case  is  simply  a  modification  of  the  general  solution, 
made  by  removing  part  of  the  mirror  about  the  point 
which  is  perpendicularly  below  0.  When  the  mirror  P  Q 
is  to  one  side,  the  spherical  waves  emitted  by  0  pass  out 
in  all  directions  ;  and  some  portions  of  them  are  inter- 
cepted by  the  surface.  The  reflected  waves  are,  then,  por- 
tions of  spheres  drawn  around  (/  as  a  centre.  The,  region, 
though,  in  which  there  are  any  reflected  waves  is  bounded 
by  the  cone  which  can  be  constructed  by  drawing  straight 


412 


THEORY  OF  PHYSICS 


[CH.  II 


lines  from  (7  to  each  point  of  the  boundary  of  the  mirror 
P  Q.  (Of  course  the  phenomena  at  the  edges  of  the 
mirror  are  complicated  by  diffraction.  See  Art.  303.)  If 
the  area  P  Q  is  very  small,  that  portion  of  the  spherical 


FIG.  230. 

waves  which  is  intercepted  by  it  is  called  a  "  pencil "  or 
"  cone  "  of  waves  ;  and  it  is  evident  that  the  action  of  the 
mirror  is  to  change  the  direction  of  the  cone  or  pencil,  so 
that  it  seems  to  come  from  the  point  0'  instead  of  from  0. 
309.  Inclined  Mirrors,  A  special  illustration  of  reflec- 
tion is  afforded  by  the  images  formed  when  a  source  of 
light  is  placed  between  two  plane  mirrors  which  are  in- 
clined to  each  other.  The  drawing  shows  the  images  in  the 
case  of  a  source  of  light,  0,  placed  between  two  plane  mirrors, 
/  and  //,  which  make  the  angle  60°  with  each  other.  The 
images  are  Of,  0" ',  P",  Pf,  P;  and  they  are  formed  in  this 
way :  Some  of  the  waves  from  0  fall  upon  mirror  /,  and  are 
reflected,  leaving  the  mirror  as  if  they  came  from  0' ;  but 
these  waves  seeming  to  come  from  Of  fall  upon  mirror 
//,  and  are  reflected,  leaving  the  mirror  as  if  they  came 


310]  REFLECTION  413 


from  P't  where  P  0'  is  a  line  perpendicular  to  mirror  // 
and  bisected  by  it ;  these  waves,  seeming  to  come  from 
P,  fall  upon  mirror  /,  and  are  reflected,  leaving  the  mir- 
ror as  if  they  came  from  P"  where  P'  P"  is  perpen- 
dicular to  mirror  /  and  bisected  by  it,  etc.  Some  of 
the  waves  from  0  also  fall  upon  the  second  mirror,  and 


have  images  P,  0" ,  P' ',  etc. ;  and  it  may  be  easily  proved 
by  geometry  that,  if  the  angle  between  the  mirrors  is 

q/?AO 

— ,  where  n  is  any  whole  number,  the  images  formed  in 

succession  in  the  two  mirrors  will  finally  coincide,  and 
that  their  number  is  n  —  1.  In  the  case  shown  in  the  fig- 
ure n  =  6,  and  there  are  5  images.  This  is  the  principle 
of  the  kaleidoscope  and  other  toys. 

310.  Geometrical  Measure  for  Curvature.  In  considering 
the  reflection  of  spherical  waves  from  spherical  surfaces,  a 
geometrical  measure  of  the  curvature  of  the  waves  and 
the  surfaces  is  often  useful.  A  spherical  surface  is  com- 
pletely defined  by  its  radius,  the  reciprocal  of  which  is 
called  the  "  curvature,"  and  it  may  be  easily  shown  that 


414 


THEORY  OF  PHYSICS 


[CH.  II 


there  is  a  simple  geometrical  method  of  representing  the 
curvature.  Let  0  be  the  centre  of  the  spherical  surface 
whose  section  by  the  paper  is  the 
circle  B  I)  A.  Draw  any  chord,  A  B, 
and  the  radius  OD  perpendicular  to 
it.  Call  B~C  =  a,  TTC  =  6 ;  and  the 
angle  B  0  D  =  0. 

Then      EG  =  a  =  r  sin  0 


FIG.  232. 


or 


C  0  =  r  —  I  =  r  cos  0 
squaring  and  adding, 

ra_2r&  +  62+  a2  =  r2 
I2  +  a2  =  2  r  b 


But,  if  the  curvature  is  very  small,  that  is,  if  r  is  very 
large  and  if  0  is  very  small,  b2  may  be  neglected  in  com- 
parison with  a2 ;  and  so 

,  .':.J  =  y:-  •  -  •:•  •  a> 

(These  conditions  are  equivalent  to  assuming  that  only  a 
small  central  portion  of 
a    spherical    surface    of 
large    radius    is    used.) 
Consequently,  if  a  is  con- 


stant,   '•-  is  proportional 


to  D  C\  or,  the  curva- 
ture of  any  sphere  is 
measured  by  D  C,  the 
"  sagitta  "  corresponding 
to  a  chord  of  fixed  length. 
Apply  this  method  to 

the     case    of     spherical  FlG  23g 

waves   incident   upon   a 

plane   surface.   .  The   curvature   of  the  incident  wave  in 
the  position    0  A  B  is   measured   by  A  M ;   that  of   the 


311] 


REFLECTION 


415 


reflected  wave  by  A'  M.  But  the  lines  A  M  and  A'  M 
are  of  equal  length,  and  in  opposite  directions.  Hence 
AM  —  —  AM\  and,  therefore,  a  plane  surface  simply 
reverses  the  curvature  of  an  incident  train  of  spherical 
waves. 

311.  Spherical  Waves  Incident  on  a  Spherical  Mirror. 
There  are  four  special  cases,  according  as  the  waves  are 
diverging  or  converging,  and  the  surfaces  concave  or 
convex.  The  same  demonstration  and  formulae,  however, 
apply  to  them  all.  The  simplest  case,  perhaps,  is  that  of 
diverging  waves  incident  upon  a  concave  surface ;  and  it 
will  be  considered  first. 

1.  Diverging  Waves,  Concave  Surface.  Let  S  be  the 
centre  of  the  concave  mirror,  and  0  be  the  source  of  di- 


FIG.  234. 

verging  spherical  waves.  Draw  a  straight  line  through  S 
and  0,  and  let  it  meet  the  surface  in  the  point  C ;  also 
draw  any  chord  perpendicular  to  the  line  through  S  and 
0.  This  line  connecting  S  and  0  is  called  the  "axis" 
of  0.  If  the  chord  of  the  sphere  meets  the  axis  in  the 
point  M,  the  curvature  of  the  surface  is  measured  by  C  M. 
Draw  a  sphere  around  0  so  as  to  have  the  same  chord  as 
that  of  the  mirror,  and  let  it  intersect  the  axis  in  A  ;  then 
the  curvature  of  the  incident  waves  at  a  certain  instant  is 


416  THEORY   OF  PHYSICS  [CH.  II 

A  M.  But,  when  the  disturbance  from  0  reaches  C,  it 
does  not  penetrate  into  the  surface  to  A,  but  is  reflected 
back  to  a  point  B  such  that  the  distance  C  B  =  A  C ;  so, 
when  the  wave  reaches  the  ends  of  the  chord,  P  and  Q,  the 
reflected  wave  has  the  wave-front  P  B  Q,  which  may  be 
proved  to  be  a  spherical  surface,  if  the  curvature  of  the 
mirror  and  the  portion  of  it  used  for  reflecting  the  waves 
are  not  too  great.  So  the  curvature  of  the  reflected  wave 
is  measured  by  B  M.  Writing  C,  Cit  and  Cr  for  the  curva- 
tures of  the  mirror,  the  incident  wave,  and  the  reflected 
wave, 

C  =  'CW,  Ci  =  ~AM,  Cr  =  B~M. 
But  AM  =  AC  +  C~M 

(remembering  that  the  line  A  Mis  the  line  from  A  to  B,  etc.) 
and  ~CM  =  ~CB 


or  BM  =  -CB  +  CM. 


Hence,  since  AC—  C  B, 


AM+  B  M  =  2  CM, 
or  Ci  +  Cr  =  2(7    .......     (2) 

In  the  figure,  as  shown,  the  reflected  wave  is  a  spherical 
w^ave  with  its  centre  on  trie-same  side  as  S  and  0.  Let  it 
be  at  Of.  Then  the  reflected  waves  will  leave  the  surface 
and  converge  towards  0' ;  so  that,  after  contracting  there  to 
a  point,  they  will  diverge  again  and  spread  out  in  ever- 
increasing  spheres.  Consequently,  the  waves  which  di- 
verge from  0  converge,  after  reflection,  at  (/  ;  and  0  and 
0'  are  called  "conjugate  foci."  Conversely,  of  course, 
waves  diverging  from  0'  will  converge,  after  reflection 
at  0. 

There  are  several  cases  of  interest. 

312.  a.  If  the  incident  waves  are  plane,  this  is  equiva- 
lent to  0  being  at  an  infinite  distance  away  on  the  axis,  and 
Ct  =  0.  Consequently  Cr  —  2  (7;  or  0'  is  half-way  be- 
tween S  and  C  at  a  point  Ft  which  is  called  the  "  principal 


313] 


REFLECTION 


417 


FIG.  235. 


focus  "  for  this  axis.  Conversely,  if  the  source  is  at  F,  the 
reflected  waves  will  be  plane  waves  whose  normal  is 
parallel  to  the  direction  of  the  axis  of  F. 

I.  If  the  source  is  at  the  centre  of  the  spherical  surface, 
i.  e.  if  0  is  at  $,  Ci  =  C .'.  Cr  =  C  \  or,  waves  diverging 
from  S  will  converge  again  to  the  same  point  after  reflection. 

313.  c.  If  an  illuminated  object,  like  an  arrow,  0  P,  is 
placed  in  front  of  a  concave  mirror,  there  will  be  a  con- 


FIG.  236. 

jugate  focus  for  each  of  its  points  on  its  corresponding 
axis  ;  and  therefore  there  will  be  an  image  of  the  entire 
arrow.  The  position  of  this  image  may  be  easily  found : 
Let  0  P  be  perpendicular  to  0  C,  the  axis  of  0.  Consider 
parallel  waves  whose  wave-normal  is  0  C  to  be  advancing 
towards  the  mirror;  they  will  reach  OP;  and  the  dis- 
turbance from  P  may  be  traced,  because  it  is  known  that 

u 


418  THEORY   OF   PHYSICS  [CH.  II 

parallel  waves  are  brought  to  a  focus  at  F,  half-way  be- 
tween S  and  C.  Consequently,  when  P  emits  spherical 
waves,  that  particular  disturbance  which  advances  along 
P  Q  parallel  to  0  C,  the  axis  of  0,  must  be  reflected  so  as 
to  pass  through  F  along  the  line  QF.  But  the  spherical 
waves  from  P  must,  after  reflection,  all  converge  to  some 
point  P  on  its  axis  ;  that  is,  P  is  the  centre  to  which  all 
the  disturbances  sent  out  from  P  must  come  after  reflec- 
tion. One  line  of  disturbances  from  P  is  reflected  in  the 
direction  Q  F\  and  therefore  P'  must  be  at  the  intersection 
of  the  axis  P  Q!  with  the  line  Q  F,  as  shown.  If  the  cur- 
vature of  the  mirror  is  not  too  great,  it  may  be  proved  that 
the  image  of  0  Pis  C/  P,  perpendicular  to  the  axis  of  0. 

314.  d.  The  position  and  size  of  the  image  may  also  be 
easily  calculated  from  the  above  formula,  d  4-  Cr  =  2  C. 
For  d  is  the  reciprocal  of  the  distance  0  A,  i.  e.  is  practi- 
cally the  reciprocal  of  0  C.  Similarly,  Cr  is  practically 
the  reciprocal  of  OF  (7;  and  (7  is  the  reciprocal  of  S  C.  Call- 
ing 0  C  =  u,  0'  C=v,  S  C  =  r,  the  above  equation  becomes 


u 


so  that,  if  r  and  u  are  known,  v  may  be  calculated. 
Further,  by  geometry,  it  is  evident  that 


OP:  a  F  =  SO:  SO'  =  u  -  r  :  r  -  v, 
and  by  equation  (2)     u  —  r  :  r  —  v  =  u  :  v. 

That  is,  the  linear  dimensions  of  object  and  image  are  in 
the  ratio  of  u  to  v.  This  ratio  of  the  linear  dimensions  is 
sometimes  called  the  linear  "  magnification." 

In  the  illustration  shown,  the  object  causes  a  real  image 
on  reflection ;  because  the  waves  actually  converge  to- 
ward 0'  P ;  and,  if  a  screen  is  placed  there,  the  image 
will  appear  on  it. 

315.  e.  In  certain  cases  the  image  may  be  virtual. 
For,  since  Ct  +  Cr  =  2  C,  if  Ct  >  2  C,  Cr  is  negative.  This 


316]  NIVERSFWCT0F  CALIFORNIA9 

means  that,  if  the  object  is  placed  nearer4  Y§ec§iirror  than 
the  principal  focus,  the  image  must  be  back  of  the  mirror, 
and  so  must  be  virtual.  A  figure  is  given  to  show  this,  in 
which  an  object  0  P  is  placed  between  F  and  C ;  and  the 
virtual  image  is  at  0'  P'.  Hence  waves  diverging  from  0 


FIG.  237. 

fall  upon  the  mirror  and  are  reflected,  seeming  to  come 
from  0' ;  that  is,  they  diverge  after  reflection  from  the 
virtual  image  O '. 

2.  Converging  Waves,  Concave  Surface.  This  is  evidently 
the  converse  of  the  case  just  considered.  For,  if  waves  are 
converging  toward  Or,  they  will  be  reflected  so  as  to  con- 
verge toward  0.  (This  is  true  whether  Of  is  a  real  or  a 
virtual  image  in  the  previous  case.) 

It  should  be  remembered  that,  if  waves  are  converging 
to  a  point  which  is  on  the  opposite  side  of  the  mirror 
from  its  centre  of  curvature  S,  the  numerical  value  of  Q 
must  be  given  a  sign  opposite  to  that  of  C. 

316.  3.  Diverging  Waves,  Convex  Surface.  Let  0  be 
the  source  of  the  waves,  and  S  the  centre  of  the  surface. 
Draw  the  axis  0  S,  and  proceed  exactly  as  in  Case  1. 


and 


C  =  CM,     Ct  =  AM,     Cr  =  BM, 
TO  =  C~B. 


420 


THEORY  OF  PHYSICS 


[CH.  II 


But 

Hence,  as  before, 


FIG.  238. 

A~M  =  A~C  +  CM 
BM  =  -CB+  CM. 


_  _ 

or,  writing         S~C  =  r,     0~C  =  u,     O~C  =  v, 
1        1    _  2 

^  "*"  v        r 

The  only  precaution  is  to  give  the  numerical  values  of  Cf 
and  Cr  and  of  u  and  v  proper  signs,  depending  upon 
whether  0  and  O  are  on  the  same  side  as  S  or  the 
opposite. 

A  special  case  is  when  Ct  <  2  C.  Then  Cr  is  positive, 
and  the  image  0'  is  on  the  same  side  of  the  surface  as  is  S. 
Consequently,  waves  diverging  from  0  will,  after  reflec- 
tion, seem  to  diverge  from  a  virtual  image  0'. 

4.  Converging  Waves,  Convex  Surface.  This  is  evidently 
the  converse  of  the  special  case  discussed  in  the  last 
section.  For,  if  waves  converge  toward  (7,  they  will, 
after  reflection,  converge  to  0. 

317.  Spherical  Aberration.  The  theory  of  spherical  mir- 
rors as  given  in  the  preceding  section  applies,  as  carefully 
noted,  only  to  small  mirrors  of  small  curvatures,  or,  what 


317] 


REFLECTION 


421 


is  the  same  thing,  to  small  portions  immediately  around 
the  axis,  if  the  mirror  is  large.  If  other  portions  of  a 
mirror  than  that  close  to  the  axis  are  used,  the  phenomena 
are  not  so  simple.  Thus,  if  spherical  waves  from  0  fall 
upon  a  large  concave  mirror,  those  waves  which  are  re- 
flected from  the  surface  near  C,  the  extremity  of  the  axis 
of  0,  are  brought  to  a  focus  at  the  point  (7  on  the  axis. 


FIG.  239. 


But  waves  which  fall  obliquely  on  the  mirror,  as  at  P  and 
Q,  are  not  reflected  to  (7,  but  illuminate  a  surface  whose 
section  by  the  paper  is  a  pointed  curve  as  shown.  This 
surface,  then,  is  rather  conical,  with  its  vertex  at  (/ ;  and 
it  is  called  the  "  caustic  "  surface.  Illustrations  are  offered 
when  light  is  reflected  by  the  sides  of  a  tumbler  or  tea- 
cup, and  are  doubtless  most  familiar.  In  general,  then, 
unless  the  waves  fall  perpendicularly  upon  a  spherical 
surface,  the  reflected  waves  are  not  brought  to  a  focus  at 
a  point,  but  illuminate  a  surface ;  the  entire  phenomenon 
being  called  "  spherical  aberration." 


CHAPTER    III 

REFRACTION 

318.  As  explained  in  the  discussion  in  SOUND  (Art.  137), 
when  waves  in  one  medium  reach  the  bounding  surface 
which  separates  it  from  another,  there  are  always  waves 
produced  in  this  second  medium.     So,  if  ether-waves  reach 
a  surface  separating  the  pure  ether  from   a  transparent 
substance,  such  as  water  or  glass,  some  waves  will  enter 
the  ether  in  which  the  matter  is  immersed.     The  same  is 
true,  of  course,  if  the  waves  reach  a  surface  which  sepa- 
rates two  transparent  substances.     The  waves  are  said  to 
be  "  refracted "  when  they  pass  from  the  pure  ether  into 
ether  in  which  matter  is  immersed,  or  from  one  trans- 
parent substance  into  another.     The  reason  for  this  name 
will  be  explained  later.     It  should  be  remembered  that  it 
has  been   proved  (Art.  301)  that  the  velocity  of  ether- 
waves  is  decreased  when  they  pass  from  pure  ether  into 
ether  loaded  with  matter.     The  wave-number  of  a  train 
of  waves  is  not  changed  by  refraction,  because  the  same 
number  of  waves  must  go  away  from  the  separating  sur- 
face as  come  up  to  it.     The  velocity  v  =  n  X ;  and,  since  v 
changes  but  n  does  not,  X,  the  wave-length,  must  change  on 
refraction. 

Several  cases  of  refracted  waves  will  be  studied  in  detail. 

319.  Plane  Waves  Refracted  at  a  Plane  Surface,    Let  plane 
waves  of  a  constant  wave-number  be  incident  upon  a  plane 
surface  separating  the  pure  ether    from  ether  containing 
matter  immersed  in  it,  e.  g.  water  or  glass.    Let  the  section 
of  the  incident  wave-front  at  any  instant  be  0  (7  O' ;  and 


319]  REFRACTION  423 

let  the  section  of  the  surface  be  0  Q  P'r,  the  line  of  inter- 
section of  the  two  surfaces  being  perpendicular  to  the 
plane  of  the  paper.  Draw  the 
wave-normal  0"  P" .  At  the 
end  of  a  certain  time,  t  sec- 
onds, the  disturbance  from  0" 
will  reach  the  surface  at  P1 ', 
where  0"  F'  =  v t  if  v  is  the 
velocity  of  the  waves  in  the 
ether.  In  this  same  time  the 
disturbance  from  0  has  spread  -p  2  .Q 

out  in  the  lower  medium  into 

a  sphere  of  radius  0  P  where  0  P  =  Vi  t,  Vi  being  the 
velocity  of^hese  particular  waves  in  the  ether  loaded  with 
the  matter.  The  wave-front  in  this  lower  medium  at  this 
time  must  include  a  line  through  P"  perpendicular  to  the 
plane  of  the  paper ;  (and  it  must  also  be  tangent  to  the 
sphere  of  radius  0  P  around  0  as  a  centre.  Draw  from  P" 
a  straight  line  tangent  to  the  circle,  which  is  the  section 
by  the  paper  of  the  sphere  around  0.  This  line  is  the 
trace  .on  the  paper  of  the  wave-front ;  for,  consider  the  dis- 
turbance from  any  point  Or  of  the  incident  wave,  which  is 
advancing  in  the  direction  0'  Q.  It  reaches  Q  when  the 
disturbance  from  0"  reaches  Q1,  if  the  line  Q  Q'  is  drawn 
parallel  to  0  0'  0" ;  and  while  this  last  disturbance  ad- 
vances to  P" ,  a  wave  will  spread  out  in  the  lower  medium 
around  Q  of  a  radius  Q  P'  where 


QP'  :  OP  =  Q' P"  :  O'  P" . 

But,  if  a  sphere  is  drawn  around  Q  tangent  to  the  line 
0"  P,  it  will  have  a  radius,  T,  which,  by  similar  triangles, 
may  be  expressed 


r  :  OP  =  P"  Q:P"0  =  Qf  P"  :  0"  P" . 


Hence  r  =  Q  P' ;  and  so  the  spherical  waves  sent  out  from 
Q  will  be  tangent  to  the  line  PP",  when  the  disturbance 


424  THEORY   OF  PHYSICS  [CH.  Ill 

from  0"  reaches  P" .  Therefore,  as  said,  the  wave-front  in 
the  lower  medium  is  a  plane  whose  section  by  the  paper 
is  the  line  P  Pf  P" ;  and  it  is  evident  that  the  direction 
of  the  wave-front  has  been  changed  on  entering  the  lower 
medium.  The  waves  are,  therefore,  said  to  be  "  refracted  " 
or  bent.  The  angle  between  the  normal  to  the  surface 
and  that  to  the  refracted  wave,  i.  e.  0  P" P,  is  called  the 
"  angle  of  refraction,"  corresponding  to  the  angle  of  inci- 
dence 0"  0  P". 
As  noted  above, 


V'P"  =  vt    and     OP  =  vrf. 
Calling  the  angle     WOP"  =  i,   and   P  P"  0  =  r, 
UrJTf  =  OT77  sin  i,     0~P  =  UP77  sin  r. 

TT  sin  i        v  ,„. 

Hence  -; —  =  — (1) 

sin  r       Vi 

v 
This  ratio,  — ,  of  the  velocity  of  waves  in  the  pure  ether  to 

that  of  certain  waves  in  ether  in  which  there  is  a  defi- 
nite kind  of  matter,  is  called  the  "  index  of  refraction  "  of 
that  matter  for  the  given  waves.  It  is  a  constant  if  the 
wave-theory  is  true.  Consequently,  the  ordinary  laws  of 
refraction  follow  at  once :  — 

1.  The  normals  to  the  incident  and  refracted  wave-fronts 
and  to    the  plane  surface    all  lie  in  one   plane,  viz.  the 
plane  of  incidence. 

2.  The  ratio  of  the  sine  of  the  angle  of  incidence  to  the 
sine  of  the  angle  of  refraction  is  a  constant  for  a  given 
form  of  matter  and  waves  of  a  definite  wave-number ;  it 
is  entirely  independent  of  the  angle  of  incidence  itself. 

These  laws  have  been  perfectly  verified  by  experiment. 

It  has  been  proved  by  direct  experiment  that,  in  all 
cases  when  the  waves  pass  from  the  pure  ether  into  ether 
contained  in  transparent  matter,  i>r;  that  is,  v  >  Vi. 
Therefore,  the  velocity  of  ether-waves  is  always  dimin- 
ished by  passing  into  such  matter. 


319]  REFRACTION  425 

It  has  been  preyed  further  by  direct  experiment  that 
waves  of  different  wave-numbers  are  refracted  differently, 
and  that  blue  light  is  refracted  more  than  green,  and  green 
more  than  red,  when  the  waves  pass  from  pure  ether  into 
any  ordinary  transparent  medium.  But,  as  proved  in 
Chapter  L,  Article  300,  red  light  is  characterized  by  hav- 
ing a  longer  wave-length  (therefore  less  wave-number) 
than  green ;  and  green  has  a  longer  wave-length  than  blue. 
Hence,  in  passing  into  ether  which  is  loaded  with  matter, 
the  longer  waves  are  not  so  much  affected  as  the  shorter 
ones,  and  are  therefore  faster.  (In  other  words,  waves  of 
small  wave-number  are  less  influenced  than  those  of  large.) 
See  Article  344. 

It  follows  from  the  above  demonstration  that,  if  the 
upper  medium  is  not  pure  ether,  but  has  matter  immersed 
in  it,  so  that  the  velocity  of  these  particular  waves  in  it 

is  vz, 

sin  iz  _  vz 

sin  ii      Vi  ' 

if  ii  is  the  angle  between  the  wave-normal  and  the  normal 
to  the  surface  in  the  medium  where  the  velocity  is  Vi ; 
and  iz  is  the  corresponding  angle  in  the  second  medium. 
If  MI  is  the  index  of  refraction  for  the  lower  medium,  and 
fjL2  for  the  upper,  i.  e.  if 


then  -  =  -  , 

Vi      M2 

sin  iz      MI  /ox 

and  - — r  —  - (*) 

sin  ii      MS 

Dr  Mi  sni  *i  —  A^  sin  ^2 (2  a) 

Some  values  of  M  f°r  different  substances  and  different 
waves  are  given  in  the  following  table.  Methods  of  exact 
measurement  will  be  discussed  later.  (It  should  be  stated 


426 


THEORY  OF  PHYSICS 


[CH.  Ill 


that  p  varies  with  the  temperature,  and   also  with  the 
pressure,  in  the  case  of  gases ;  but  the  variations  are  slight.) 


TABLE  XVI 
INDICES  OF  REFRACTION 


Subsfance. 

Wave-length, 
cm. 

Index. 

Temperature. 
p  =  76  cm. 

Air            ..... 

0.0000589 

1.0002922 

0°  C 

t  . 

0.0000485 

1.0002943 

0° 

a 

0.0000434 

1.0002962 

0° 

0.0000589 

1.000043 

Hydrogen      .... 
Nitrogen  

0.0000589 
0.0000589 

1.000140 
1.000297 

0° 
0° 

0.0000589 

1.000272 

0° 

Alcohol     

0.0000589 

1.360 

15° 

Chloroform    .... 

0.0000589 

1.449 

15° 

Carhon  Bisulphide 

a             t( 

Water       

0.0000589 
0.0000485 
0.0000589 

1.624 
1.648 
1.334 

25° 
25° 
16° 

« 

0.0000485 

1.338 

16° 

ti 

0.0000434 

1.341 

16° 

Rock  Salt      .... 

0.0000589 

1.5441 

24° 

((            H 

0.0000485 

1.5531 

24° 

(I          It 

0.0000434 

1.5607 

'24° 

Flint  Glass  .... 

0.0000589 

1.651 

((       a 

0.0000485 

1.665 

u      n 

0.0000434 

1.677 

Crown  Glass       .     . 

0.0000589 

1.517 

it                 U 

0.0000485 

1.524 

u            n 

0.0000434 

1.529 

Several  special  cases  of  the  refraction  of  plane  waves  at 
a  plane  surface  are  of  interest. 


320] 


REFRACTION 


427 


320.   o.  Total  Reflection. 

incidence  and  refraction  is 


The  formula  for  the  angles  of 


sin  ii  = 


sin  ^2, 


where 


Assume  that  v2  is  greater  than  vi,  and  that  the  waves  are 

incident  from  the  medium  in  which  the  velocity  is  less. 

i2    is,    then,     greater 

than  ij  ;  and,  if  PI  0 

and  0  P2  are  the  wave- 

normals    in    the    two 

media,  iz  and  ii  will 

be  as  shown.     iz  can- 

not   be    greater   than 

90°  ;  and  so  the  great- 

est value  which  i\  can 

have    if    a    refracted 

wave  is  still  produced 

is  that  found  by  sub- 

stituting for  iz  90°. 


FIG.  241. 


Hence  this  maximum  value  of  ily 


which  may  be  called  a,  is  given  by  the  equation 


/ii  sin  a  =  i*2 (3) 

(for  sin  90°  =  1).  If  ii  is  increased  beyond  a,  no  waves 
can  be  refracted  ;  and  they  will  all  be  reflected,  as  shown. 
This  angle  a,  which  marks  the  limiting  value  of  ^  before 
total  reflection  follows,  is  called  the  "  critical "  angle.  It 
is  observed,  of  course,  only  when  waves  are  passing  from 
one  medium  into  another  in  which  the  velocity  is  greater, 
e.  g.  from  water  into  air. 

As  already  noted,  p  is  different  for  waves  of  different 
wave-numbers  ;  and  so  it  would  be  expected  that  different 
waves  would  have  different  critical  angles,  which  is  an 
observed  fact.  This  determination  of  the  critical  angle 
may  be  made  quite  exactly  ;  and  so  this  is  one  method  of 
measuring  indices  of  refraction,  for 


428 


THEORY  OF  PHYSICS 


[CH.  Ill 


-  =  sn  a 


(3  a) 


321.   b.  Refraction  through  a  Plate  with  Plane  Parallel 
Faces.     When  plane  waves  fall  upon  a  plane  surface,  the 

connection  between  the  angles  of 
incidence  and  refraction  is  given 


sn   i  = 


sn 


^^ 


if  ii  is  the  angle  between  the 
wave-normal  and  normal  to  the 
surface  in  the  first  medium,  and 
iz  the  corresponding  angle  in  the 
second  medium.  If  the  second 
medium  is  bounded  by  two  par- 
allel surfaces,  the  refracted  waves 
will  fall  upon  its  second  surface 

at  an  angle  i2.  Therefore,  calling  i  the  angle  of  refraction 
of  these  waves  when  they  emerge  again  into  the  first 
medium  on  the  other  side  of  the  plate, 


FIG.  242. 


sn    2  = 


sn 


Hence  i  =  ii',  and  it  follows  that  the  waves  emerge  par- 
allel to  the  direction  which  they  had  before  they  entered 
the  plate.  A  plate  with  plane  parallel  sides  does  not,  then, 
turn  the  direction  of  plane  waves.  But,  if  the  path  of  the 
disturbance  due  to  any  point  in  the  incident  waves  is  traced, 
it  will  be  found  to  be  shifted  one  side  parallel  to  itself. 
Thus,  the  disturbance  from  0  travels  to  0'  to  0"  to  0"'  ; 
but  the  line  0  O  t  although  parallel  to  the  line  0"  0"',  is 
not  coincident  with  it,  as  is  seen  from  the  figure.  If 
ii  =  o,  however,  the  two  lines  do  coincide.  It  follows 
from  this  that,  if  the  incidence  upon  the  plate  is  normal, 
there  is  no  change  in  the  path  of  the  waves  ;  but,  if  the 
incidence  is  oblique,  the  emerging  waves,  although  parallel 
to  the  incident  ones,  are,  as  it  were,  shifted  one  side. 


322] 


REFRACTION 


429 


322.  c.  Refraction  through  a  Prism.  A  prism  is  a  piece 
of  transparent  substance,  two  portions  of  whose  bounding 
surface  are  planes,  inclined  to  each  other.  If  these  planes 
actually  intersect  in  a  line,  it  is  known  as  the  "  edge  "  of 
the  prism,  and  in  any  case  these  two  plane  surfaces  are 
called  the  "  faces  "  of  the  prism ;  and  the  angle  which  they 
make  with  each  other  is  called  the  "  angle  of  the  prism." 
If  plane  waves  fall  upon 
one  of  the  faces  of  a 
prism  so  that  the  plane 
of  incidence  is  perpen- 
dicular to  the  edge  of 
the  prism,  the  refracted 
and  emerging  waves  will 
be  as  shown  in  the  fig-  FIG  243 

ure ;   and  it   is  obvious 

that  the  effect  of  the  prism  is  to  change  the  direction  of 
the  waves.  But,  since  the  refraction  is  different  for  waves 
of  different  wave-numbers,  it  would  be  expected  that  a 
prism  would  deviate  various  waves  differently,  —  a  fact 
which  is  actually  observed.  (It  ought  to  be  stated  that 
it  is  by  means  of  a  prism  that  this  fact  in  regard  to  the 
refraction  being  different  for  different  waves  is  most  easily 
proved.) 


FIG.  244. 


The  "  deviation "  of  a  given  prism  for  a  given  train  of 
plane  waves  is  the  change  in  direction  produced  by  it  in 


430      .  THEORY  OF  PHYSICS  [CH.  Ill 

the  waves.  Its  value  may  be  easily  calculated  in  terms  of 
measurable  quantities.  Let  the  directions  of  the  wave- 
normals  and  the  normals  to  the  faces  of  the  prism  be  as 
shown  in  the  figure.  Call  the  angle  of  incidence  i  ;  the 
angle  of  refraction  r  ;  the  angle  of  incidence  on  the  sec- 
ond face  of  the  prism  /  ;  the  angle  of  emergence  i'.  If 
the  incident  wave-normal  is  prolonged  until  it  meets  the 
wave-normal  of  the  emerging  waves  prolonged  backward, 
as  shown,  8  is  the  deviation.  Call  the  angle  of  the  prism 
A.  Then,  from  ordinary  geometry,  it  follows  that 


} 

=  (i  -  r)  +  (ir  -  /)  =  i  +  i'  -  A     ' 

So  it  is  evident  that  S  depends  upon  the  angle  of  incidence, 
the  angle  of  the  prism,  and  the  index  of  refraction  (because 
i'  depends  upon  //,). 

It  may  be  proved  theoretically,  as  well  as  by  experiment, 
that  for  any  definite  train  of  waves  there  is  a  certain  mini- 
mum value  of  the  deviation,  smaller  than  which  it  cannot 
become  for  a  given  prism  ;  and  that,  further,  this  minimum 
deviation  occurs  when  the  incident  and  emerging  waves  are 
symmetrical  with  reference  to  the  prism,  i.  e.  when  i  =  i'. 

-D    *.    Sm  *  J    sm  *'  1.  -£    •          -r  rr 

But  —  -  =  fji,  and  —  -  -.  =  u,  ;  hence,  if  i  =  i  ,  r  =  r'.     If, 
sin  r  sm  / 

then,  D  is  the  angle  of  minimum  deviation  and  i  and  r  the 
corresponding  values  of  those  angles, 

A  =  2  r,  . 


This  gives  at  once  a  method  for  the  determination  of  ft  ;  for 

A  +  D 

.     .      sin  —  -  — 
sm  i  2 


.    A 
sm- 


and  A  and  D  can  both  be  easily  measured. 


323] 


REFRACTION 


431 


323.   Spherical  Waves  Refracted  at  a  Plane  Surface.     The 

character  of  the  refracted  waves  may  be  easily  determined 
for  this  case,  if  the  definition  and  measure  of  curvature 
given  in  Article  310  is  remembered.  As  the  simplest 
illustration,  consider  spherical  waves  diverging  from  a 
point  0  in  one  medium ;  and  draw  a  line  0  M  perpen- 
'  dicular  to  the  plane  surface  separating  the  two  media.  If 


FIG.  245. 


the  surface  was  not  there,  the  curvature  of  the  incident 
waves  at  a  certain  instant  when  the  radius  was  OA 
would  be  A  M\  but,  owing  to  the  presence  of  the  second 
medium,  the  disturbance  which  would  have  gone  from  M 
to  A  really  goes  from  M  to  B  (say),  so  that  the  section  of 
the  refracted  wave  is  a  circle  passing  through  B,  and  the 
ends  of  the  chord  in  which  the  circle  of  radius  0  A  around 
0  cuts  the  plane  surface.  The  centre  of  the  refracted 
waves,  i.  e.  the  image  of  0,  is  a  point  #'  on  the  line  0  M\ 
and  the  curvature  of  the  refracted  wave  at  this  instant  is 
BM  where  B  M :  AM=v2:  Vi,  the  ratio  of  the  veloci- 
ties in  the  two  media. 

Hence,  writing  Cy  and  Or  for  the  curvature  of  the  inci- 
dent and  refracted  waves, 


or 


(6) 


432  THEORY  OF  PHYSICS  [CH.  Ill 

Therefore  the  refracted  waves  diverge  as  if  from  a  centre 
(/  where  Of  B  =  1  /  Cr.  (Conversely,  waves  in  the  second 
medium  converging  toward  a  point  (/  in  the  first  will  con- 
verge, after  refraction,  at  the  point  0.)  Expressed  in  terms 

of  distances, =  —  ;  or,  very  approximately, 

OB       /*2 


324.  A  special  application  of  this  formula  furnishes  a 
method  for  the  determination  of  /*i  /  ^  If  there  is  a 
source  of  light  at  0,  a  point  on  the  under  surface  of  a 

plate  of  a  substance  which  has 

parallel  sides,  and  whose  thick- 
ness is  d,  the  waves,  diverging 
from  0  and  refracted  out  from 


0' 


O  the  upper  surface  into  the  sur- 

FIG.  246.  ,.  .,, 

rounding  medium,  will  seem  to 

diverge  from  another  point,  0 ' .  (The  drawing  is  made 
so  as  to  apply  to  a  plate  of  glass  surrounded  by  air, 
or  to  a  plate  of  any  transparent  substance  surrounded  by 

a  medium  in  which  the  velocity  is  greater.     -=^  =  —  , 

OM       /** 

where  ^  is  the  index  of  refraction  for  the  plate,  and  /*2 
for  the  surrounding  medium.  0  M  is  the  thickness  of  the 
plate,  d ;  and,  as  0  Of  can  be  easily  measured,  (7  M  may 
be  calculated,  and  thus  y^i  /  /-t2  determined.  The  simplest 
method  to  measure  0  O  is  to  focus  a  vertical  microscope  on 
the  point  0  when  the  plate  is  not  in  place,  then  to  interpose 
the  plate  horizontally,  and  measure  the  distance  through 
which  it  is  necessary  to  move  the  microscope  in  order 
again  to  see  the  object,  which  is  really  at  0  but  which 
seems  to  be  at  (7. 

If  the  plate  has  a  refractive  index  greater  than  that  of 
the  surrounding  medium,  i.e.  if  ^i  > /x2,  OM>  0' M,  or 
the  point  0'  is  raised  up  nearer  the  upper  surface  of  the 


325] 


REFRACTION 


433 


plate.  This  explains  the  fact  so  often  observed,  that  an 
object  at  the  bottom  of  a  vessel  of  water  never  seems  to 
be  so  far  below  the  surface  as  the  known  depth  of  the 
water. 

325.  Another  application  is  to  spherical  waves  falling 
upon  a  prism.  Let  the  spherical  waves  diverge  from  0; 
after  entering  the  prism  they  will  seem  to  diverge  from  a 
point  (?,  where  the  line  0  0?  is  perpendicular  to  the  first 


FIG.  247. 

face  of  the  prism.  On  emerging  from  the  prism  the 
waves  diverging  from  (7  will  seem  to  diverge  from  0" 
where  the  line  (7  0"  is  perpendicular  to  the  second  face 
of  the  prism.  Thus  the  waves  which  originally  diverged 
from  0  leave  the  prism  as  if  they  diverged  from  0",  so 
that  0"  is  a  virtual  image  of  0.  Since  the  refraction 
varies  for  waves  of  different  wave-numbers,  i.  e.  for  waves 
of  different  colors  if  they  are  light-waves,  each  train  of 
waves  of  a  definite  wave-number  will  have  an  image  of  its 
own.  If,  then,  waves  of  different  wave-numbers  are  emitted 
from  0,  there  will  be  a  series  of  corresponding  virtual 
images  0" ;  and  the  emerging  waves  will  not  all  diverge 
from  the  same  point.  This  fact  is  said  to  be  due  to 
"  chromatic  aberration,"  because  different  colors  have  differ- 
ent images.  (It  should  be  noticed  that  the  waves  do  not 


434  THEORY   OF  PHYSICS  [CH.  Ill 

fall  normally  on  both  surfaces  ;  and  consequently  there  will 
be  spherical  aberration  in  general,  and  the  waves  will  not 
diverge  from  a  point  0" \  but  from  a  surface.  See  Article 
317.) 

It  is  of  interest  in  this  case  to  study  the  changes  pro- 
duced by  the  prism  in  a  "  pencil "  or  cone  of  homogeneous 
waves  leaving  0.  Consider  any  pencil  leaving  0.  On 


FIG.  248. 

entering  the  prism,  it  will  seem  to  come  from  O',  and  its 
path  will  be  as  shown.  On  leaving  the  prism,  it  will  seem 
to  come  from  0",  as  shown ;  and  it  is  evident  that  the 
effect  of  the  prism  is  to  alter  the  direction  of  the  pencil. 
(It  may  be  proved  that,  if  the  pencil  is  very  small,  and  if 
it  falls  upon  the  prism  so  as  to  have  minimum  deviation, 
it  will  leave  the  prism  as  if  it  came  from  a  point  0".  In 
any  other  case  than  this  the  pencil  will  not  seem  to  come 
from  a  point,  but  will  be  more  complicated  owing  to 
spherical  aberration.) 

326,  Plane  Waves  Refracted  at  Spherical  Surfaces.  The 
simplest  case  is  when  the  surface  is  convex.  Let  £be  the 
centre  of  the  spherical  surface ;  and  draw  from  it  a  line 
S  0  perpendicular  to  the  incident  plane  waves.  This  is 
called  the  "  axis."  Draw  an  arbitrary  chord  for  the  section 
of  the  spherical  surface  by  the  paper ;  and  let  it  be  perpen- 
dicular to  the  axis,  which  it  intersects  in  the  point  M. 
Then  the  curvature  of  the  surface  is  C  =  C  M.  When  the 


326] 


REFRACTION 


435 


waves  enter  the  surface,  they  cease  to  be  plane,  because 
in  the  time  during  which  the  disturbance  would  have  gone 


FIG.  249. 


from  C  to  M  in  the  first  medium,  it  goes  a  different  dis- 
tance, C  B,  where 


CB 


The  refracted  wave-front  is  a  sphere  whose  section  by 
the  paper  is  the  circle  through  the  point  B  and  the  ends 
of  the  fixed  chord ;  and  the  waves  are  converging  toward 
some  point  beyond  the  centre  S.  Their  curvature  at  the 
surface  is  B  M.  That  is, 


=  BM=  CM-  CB  = 


Hence 


(8) 


This  same  formula  applies  equally  well  to  plane  waves 
incident  upon  a  concave  surface.  Let  the  surface  have  a 
curvature  C  =  A  M.  When  the  plane  waves  reach  it,  the 
central  disturbance  along  M  goes  to  A  ;  but  at  the  ends 
of  the  chord  of  fixed  length  the  disturbances  enter  the 
second  medium  a  distance  M  B,  where 


436 


THEORY  OF  PHYSICS 

4 

o'        s 


[CH.  Ill 


FIG.  250. 
BM        v 


AM        vi      P"> 
and  the  refracted  wave  has  the  curvature 


Hence 


In  both  of  these  cases  of  refraction  there  is  chromatic 
aberration  if  waves  of  different  wave-numbers  are  incident 
upon  the  surfaces,  because  pi  and  /x2  are  different  for  dif- 
ferent waves,  and  therefore  Cr  varies.  There  will  also  be 
spherical  aberration,  unless  only  a  small  portion  of  the 
spherical  surfaces  immediately  around  the  axis  is  used. 

These  illustrations  in  this  article  and  in  Article  323 
are,  of  course,  special  cases  of  the  general  problem  of 
spherical  waves  refracted  at  spherical  surfaces,  because  a 
plane  is  but  a  sphere  whose  radius  is  infinite,  i.  e.  whose 
curvature  is  zero. 

327.  Spherical  Waves  Refracted  at  a  Spherical  Surface, 
There  are  four  special  cases  according  as  the  waves  are 
diverging  or  converging,  and  the  surfaces  concave  or  con- 
vex. There  is  one  general  formula  which  applies  to  all, 
however  ;  and  it  will  be  deduced  for  the  case  of  diverging 
waves  refracted  at  a  concave  surface,  its  detailed  proof  for 


327] 


REFRACTION 


437 


the  other  cases  being  left  as  an  exercise  to  the  student, 
although  the  figures  will  be  given. 

1.  Diverging  Waves  Refracted  at  a  Concave  Surface. 
Let  S  be  the  centre  of  the  concave  surface,  and  0  the 
source  of  the  diverging  waves.  Draw  a  line  through  S 
and  0,  and  let  it  meet  the  surface  at  the  point  C.  This 
line  is  called  the  axis  of  0.  Draw  an  arbitrary  chord  per- 
pendicular to  the  axis,  and  let  it  cut  it  in  the  point  M. 
Draw  a  circle  around  the  point  0,  so  that  it  passes  through 


FIG.  251. 

the  eno\s  of  this  chord ;  and  let  it  cut  the  axis  in  the  point 
A.  When  the  disturbance^  from  0  reaches  the  surface  at 
C,  it  advances  a  distance  OB  in  the  second  medium  in  the 
time  during  which  it  would  have  advanced  C  A  in  the  first 
medium.  So  the  refracted  wave-front  passes  through  B 
and  the  ends  of  the  fixed  chord ;  and  its  centre  is  0' . 
Therefore,  calling  C,  Cit  Cr  the  curvatures  of  the  surface 
the  incident  wave  and  the  refracted  wave, 


C  =  CM,   C,  =  AM,    Cr  ==  BM, 


and 


BC 


AC 
Call  this  ratio  ^  /  IJLZ,  b. 


438 


THEORY  OF  PHYSICS 


[CH.  Ill 


By  geometry 
Therefore 


BM=BC+  CM, 

AM=AC+  CM. 


Ct=C+     AC. 

Cr  =  b  C>  +  C(l-b)    .     .     . 


Hence 

2.  Diverging  waves  refracted  at  a  convex  surface. 


.     (9) 


C  =  CM,   Ci  =  AM,    Cr  =  BM, 
~B~C        v,      >i 


AC 


C, 
AM=.^LC+  CM. 


Hence 


FIG.  252. 
3.   Converging  waves  refracted  at  a  concave  surface. 


C=CB,    Ci  =  AB,    Cr=CD, 


BD, 


~AG 


Hence 


(7(1-6). 


327] 


REFRACTION 


439 


FIG.  253. 
4.  Converging  waves  refracted  at  a  concave  surface, 

c  —  We  C-  =  A^Q  c  =  WD 
(To   _  v2  _  MI  _ 

~AB  ~vi~~f*2~ 


Hence 


BD  =  BC  +  CD, 
AC  =  ~BC  +  AB. 

Cr  =  bC, 


FIG.  254. 


440  THEORY  OF  PHYSICS  [CH.  Ill 

The  same  formula  applies  to  all  four  cases,  if  it  is  re- 
membered that  the  numerical  values  of  Ct-  and  Cr  are  posi- 
tive if  0  and  0'  lie  on  the  same  side  of  the  surface  as  S, 
and  negative  if  they  lie  on  the  opposite  side. 

The  physical  description  of  the  phenomenon  is  that  inci- 
dent spherical  waves  whose  centre  is  0  are  changed  by 
refraction  at  a  spherical  surface  into  spherical  waves  with 
their  centre  at  0' ;  their  curvature  is  changed.  (7  is  the 
image  of  0 ;  and  it  may  be  real  or  virtual. 

In  all  these  cases  there  will,  of  course,  be  chromatic 
aberration  if  waves  of  different  wave-numbers  are  divem- 

O 

ing  from  or  converging  to  the  same  source  ;  and  there 
will  be  spherical  aberration  unless  the  waves  are  incident 
upon  the  surfaces  immediately  around  the  axis. 

Lenses.  A  lens  is  a  piece  of  transparent  substance, 
the  two  principal  portions  of  whose  bounding  surface 
are  curved  surfaces,  usually  spherical  (one  surface  may 
be  plane).  There  are  many  types,  depending  upon  what 
kind  of  surfaces  are  combined,  —  concave  with  concave, 
or  with  convex;  the  concavities  or  convexities  being 
in  the  same  or  opposite  directions.  Two  cases  will  be 
treated  here,  —  a  double  concave  lens  and  a  double  con- 
vex one,  and  in  both  cases  the  lens  will  be  considered 
extremely  thin. 


328.   1.  Double  concave  lens,  as  shown.     Let  S  and  S'  be 
the   centres   of    the    two    spherical    surfaces.      The    line 


328]  REFRACTION  441 

joining  them  is  called  the  "axis"  of  the  lens;  and  the 
point  where  the  axis  cuts  the  lens,  if  the  lens  is  ex- 
tremely thin,  is  called  the  "  centre  "  of  the  lens.  Con- 
sider the  case  of  spherical  waves  diverging  from  a  point  0 
on  the  axis  and  on  the  same  side  of  the  lens  as  is  S.  Call 
the  surface  whose  centre  is  S  the  first  surface,  and  the 
other  the  second.  Let  the  curvature  of  the  first  surface 
be  C,  and  that  of  the  second  C'.  Let  the  incident  waves 
have  a  curvature  0*.  Then,  by  formula  (9),  the  curvature 
of  the  refracted  waves  is 

tfr  =  &<?.+  (7(1-6). 

If  the  lens  is  very  thin,  the  waves  which  enter  at  the 
first  surface  will  reach  the  second  before  their  curvature 
has  changed  ;  so  the  curvature  of  the  waves  which  are 
incident  on  the  second  surface  from  within  has  the  nu- 
merical value  Cr  ;  but  its  sign  is  negative,  because  a 
curvature  which  is  positive  for  the  first  surface  is  negative 
for  the  second,  since  their  curvatures  are  turned  in  oppo- 
site directions.  Therefore  the  waves  incident  on  the  sec- 
ond surface  have  a  curvature  —  Cr,  when  referred  to  it. 
In  emerging  from  the  lens  into  the  surrounding  medium 
the  formula  may  be  written 

C  =  I'  C!  +  C'  (1  -  V), 


in  which,  as  just  explained,  (?/  =  —  Gr     Further,  since 

V  =  ^  ,  and  I  =  ^  ,  V  =  \.     Therefore   the  curvature  of 
Mi  ^2  b 

the  emerging  waves  is  given  by  the  formula 


(b  -  l)-r        .....     (10) 


442 


THEORY   OF  PHYSICS 


[CH.  Ill 


C  4-  C 
Now,  — 7 —  is  an  essentially  positive  quantity;  and,  if 

the  lens  has  a  greater  refractive  index  than  the  surround- 
ing medium,  b  is  less  than  1 ;  and  so  b  —  1  is  negative. 

C  4-  C' 
Therefore   the  quantity   (b  —  1)  — is  an  essentially 

negative  constant  for  any  definite  train  of  waves  for  which 
Pi  I  fj,2  is  constant,  i.  e.  for  a  train  of  waves  of  definite 
wave-number.  Write  for  it  Cf.  Then  the  above  equation 
becomes 

Ce+  d=  Cf    .     .     ...     .    ..     (10  a) 

Ce  is  positive  if  the  centre  of  curvature  of  the  emerging 
waves  is  on  the  same  side  of  the  lens  as  the  centre  of  the 
second  surface,  S1.  Similarly,  d  is  positive  if  the  centre 
of  the  incident  waves  is  on  the  same  side  of  the  lens  as  S, 
the  centre  of  the  first  surface.  It  must  be  noted  that  Cf  is 
different  for  different  values  of  ^  /  /j,2,  i.  e.  for  waves  of  dif- 
ferent wave-numbers  ;  and  consequently  lenses  have  chro- 
matic aberration. 

Several  special  cases  may  be  of  interest.  (In  the  figures 
a  thick  line  represents  the  lenses,  simply  in  order  to  help 
the  clearness  of  the  drawing.) 


FIG.  256. 

329.  a.  Incident  waves  are  plane,  and  their  normal  is 
parallel  to  the  axis  of  the  lens;  i.  e.  d  =  0  .'.  Ce  =  Cf. 
Cf  is  essentially  negative,  and  therefore  the  emerging  waves 


331]  REFRACTION  443 

diverge  from  a  point  0'  on  the  axis  on  the  opposite  side  of 
the  lens  from  S'.  This  point,  0 ,  is  called  the  "  principal 
focus  "  on  that  side  of  the  lens.  (Different  trains  of  waves 
will  have  different  principal  foci,  depending  upon  the  value 
of  their  wave-numbers,  i.  e.  upon  ^  /  ^2.)  The  distance 
of  this  point  from  the  lens  is  1  /  Cf)  and  it  is  called  the 
"  focal  distance."  There  is,  of  course,  another  principal 
focus  on  the  opposite  side  of  the  lens  at  an  equal  distance 
from  the  lens.  The  effect  of  the  lens  is,  then,  to  make 
plane  waves  diverge  as  if  from  a  centre  at  the  principal 
focus ;  and  for  this  reason  such  a  lens  is  called  a  "  diverg- 
ing lens." 

A  disturbance,  therefore,  starting  from  any  point  of  a 
wave-front  and  advancing  parallel  to  the  axis  will  emerge 
from  the  lens  as  if  it  came  from  the  principal  focus  on  the 
opposite  side. 

330.  I.  Incident  waves  are  converging  on  one  side  to- 
wards the  principal  focus  on  the  opposite  side ;  i.  e.  (7,-  == 
Cj .'.  Ce  —  ^.     Hence  the  emerging  waves  are  plane;  and 
their  wave-normal  is  parallel  to  the  axis. 

It  follows  from  this  that  a  disturbance  apparently  pointed 
toward  the  focus  on  the  other  side  of  the  lens  will  be 
turned  by  refraction  so  as  to  emerge  parallel  to  the  axis. 

This  case  is  evidently  just  the  reverse  of  case  a,  as  is 
apparent  from  the  diagram. 

331.  c.  Corresponding  to  any  centre^  of  incident  waves, 
there  will,  of  course,  be  a  centre  of  emerging  waves,  if  the 
lens  is  bounded  by  surfaces  of  small  curvatures,  and  if 
only  the  central  portions   of  them  are  used.      Therefore 
there  will  be  an  image  of  any  object ;  and  it  is  not  diffi- 
cult to  construct  it  by  graphical  methods.     Let  F  and  F 
be  the  principal  foci  on  the  two  sides  of  the  lens  ;  and  let 
the  illuminated  object  0  P  be  placed  at  0,  perpendicular  to 
the  axis.     P  sends  out  spherical  waves,  i.  e.  series  of  dis- 
turbances  in   all    directions.      The    disturbance    sent  out 
parallel  to  the  axis  will  emerge  as  if  it  came  from  F,  the 


444 


THEORY  OF  PHYSICS 


[CH.  Ill 


principal  focus  on  the  same  side  of  the  lens  as  P.  The 
disturbance  directed  toward  F't  the  other  principal  focus 
on  the  opposite  side,  emerges  from  the  lens  parallel  to 
the  axis.  Therefore  the  image  of  P,  i.  e.  the  centre  of  all 
the  emerging  disturbances,  must  be  the  point  P,  the  point, 
of  intersection  of  the  two  lines  as  drawn.  Similarly,  the 
image  of  0  will  be  at  0' ;  and  the  object  0  P  has  a  virtual 


FIG.  257. 

image  at  (/  P.  The  lens  is  supposed  to  be  extremely  thin, 
and  a  disturbance  sent  from  P  directly  to  the  centre,  C,  of 
the  lens,  i.  e.  the  point  where  the  axis  cuts  it,  will  pass 
through,  keeping  its  direction  unchanged,  and  therefore  P, 
the  image  of  P,  must  lie  on  this  line  also. 

The  "  linear  magnification,"  or  the  ratio  0'  P  /  6TP,  may 
be  determined  by  means  of  similar  triangles. 


0'  P'      0'  C 


OP 


i 

Cl 


c.m 


332.   d.  The  general  formula  may  be  expressed  in  terms 
of  distances.     For,  in  the  equation 

CJ  +  C.  =  C,  , 
<7.  =  J-  =  i.    c  -  -1     !-    r       1       i 

OC~  ~='       f~==' 


u 


333]  REFRACTION  445 

Hence  ^+^  =  7; '(U) 

in  which  formula  u  is  positive  if  the  object  is  on  the  same 
side  of  surface  /  as  its  centre ;  v  is  positive  if  the  image  is 
on  the  same  side  of  surface  77  as  its  centre  ;  and  /  is  an 
essentially  negative  quantity  (in  all  ordinary  lenses,  such 
as  glass  surrounded  by  air),  whose  numerical  value  may 
be  determined  by  experiment. 

333.   2.  Double  Convex  Lens,  as  shown.     The  centre  of 
the  first  surface  is  now  on  the  opposite  side  of  the  lens 


s' 


FIG.  258. 


from  the  surface  itself  ;  and  the  same  is  true  of  the  centre 
of  the  second  surface.  The  formula  as  deduced  for  the 
double  concave  lens  applies  to  this  case  also,  because  all 
of  the  equations  are  perfectly  general,  — 


where  Ct  is  positive,  if  the  centre  of  the  incident  waves  is 
on  the  same  side  of  the  first  surface  as  its  centre  ;  Ce  is 
positive  if  the  centre  of  the  emerging  waves  is  on  the  same 
side  of  the  second  surface  as  its  centre  ;  Cf  is  essentially 
negative  if  the  lens  is  made  of  a  material  whose  index  of 
refraction  is  greater  than  that  of  the  surrounding  medium. 
In  general,  the  centre  of  the  incident  waves  will  not  be  on 


446 


THEORY  OF  PHYSICS 


[CH.  Ill 


the  same  side  of  surface  /  as  is  its  centre  S ;  and  it  will 
obviously  be  more  convenient  to  measure  the  distance  of 
the  centre  of  the  incident  waves  from  the  surface  /  as 
positive,  if  this  centre  is  on  the  opposite  side  from  $,  the 
centre  of  the  surface  /.  Similarly,  it  is  in  general  more 
convenient  to  measure  Ce  as  positive  if  the  centre  of  emerg- 
ing waves  is  on  the  opposite  side  of  surface  //  from  its 
centre  S'.  Therefore,  if  the  sign  of  each  term  in  the  gen- 
eral formula  is  changed,  it  may  be  written  the  same  as 
before, 

c  +  c.=  c,, 

where  Ct  is  positive  if  the  centre  of  the  incident  waves  is 
on  the  same  side  of  the  lens  as  S' ;  Ce  is  positive  if  the 
centre  of  the  emerging  waves  is  on  the  same  side  of  the 
lens  as  S ;  and  Cf  is  now  an  essentially  positive  constant, 
if  the  lens  has  a  greater  refractive  index  than  the  sur- 
rounding medium,  and  if  the  waves  have  a  definite  wave- 
number.  Its  numerical  value  will  change  with  the  wave- 
number. 

Some  special  cases  may  be  discussed. 

334.  a.  Incident  waves  are  plane,  and  their  wave-normal 
is  parallel  to  the  axis  ;  i.  e.  d  =  0  .*.  Ce  =  C .  Therefore, 


FIG.  259. 


since  Cf  is  positive,  the  curvature  Ce  is  positive ;  and  the 
centre  of  the  emerging  waves  is  at  a  point  on  the  axis  on 
the  opposite  side  of  the  lens  from  the  incident  waves,  at 
a  distance-  from  the  lens  I/O,  =/.  This  point  and  the 


336] 


REFRACTION 


447 


corresponding  one  at  an  equal  distance  on  the  opposite 
side  of  the  lens  are  called,  as  before,  the  principal  foci ; 
and,  owing  to  chromatic  aberration,  waves  of  different 
wave-numbers  will  have  different  foci.  The  effect  of  the 
lens  is  thus  seen  to  be  to  converge  the  waves  to  a  focus ; 
and  so  it  is  called  a  "  converging  lens."  It  follows,  too,  that 
a  disturbance  propagated  parallel  to  the  axis  will  emerge 
in  such  a  direction  as  to  pass  through  the  principal  focus. 

335.  ~b.  Incident  waves  are  diverging  from  a  principal 
focus,  i.  e.  d=  Of.'.  Ce  =  o.     Hence  the  emerging  waves 
are  plane,  with  their  wave-normal  parallel  to  the  axis.     A 
disturbance,  then,  propagated  through  a  principal   focus 
and  incident  on  the  lens  will  emerge  in  a  direction  parallel 
to  the  axis. 

This  case  is  obviously  just  the  reverse  of  case  a,  as  will 
be  seen  on  reference  to  the  figure. 

336.  c.  Corresponding  to  any  centre  of  incident  waves 
there   is    (in  general)  a   centre  of  emerging  waves ;  and 


FIG.  260. 

therefore  there  will  be  an  image  of  any  object.  Let  F 
and  Fr  be  the  two  principal  foci,  and  let  the  object  0  P 
be  placed  perpendicular  to  the  axis  at  a  point  beyond  the 
focus  F.  The  point  P  is  sending  out  spherical  waves  ; 
and  the  disturbance  parallel  to  the  axis  will  emerge  from 
the  lens  in  such  a  direction  as  to  pass  through  the  prin- 
cipal focus  Ff ;  the  disturbance  in  the  direction  through 
the  principal  focus  F  on  the  same  side  of  the  lens  as  P, 
i.  e.  in  the  direction  PF,  will  emerge  from  the  lens  par- 


448 


THEORY  OF  PHYSICS 


[CH.  Ill 


allel  to  the  axis.  Consequently  the  centre  of  the  emerg- 
ing waves  is  the  point  of  intersection  of  these  lines  of 
disturbances,  F.  The  waves  from  P  will  actually  con- 
verge there  and  then  diverge  again,  so  that  Pf  is  a  real 
image.  The  lens  is  supposed  very  thin  ;  so  a  disturb- 
ance from  P  to  the  centre  of  the  lens,  the  point  C,  will 
pass  through,  keeping  its  direction  unchanged.  P  must, 
therefore,  also  lie  on  the  line  P  C. 

The  image  of  0  will  be  at  0',  where  (7  P  is  perpendicu- 
lar to  the  axis  ;  and  the  image  of  0  P  is  real  and  inverted. 

The  linear  magnification 


O'P      CO' 


OP       CO 


c_ 

~0e' 


337.   d.  Another  special  case  would  be  when  the  object 
is  placed  between  the  principal  focus  and  the  lens,  the 


FIG.  261. 

graphical  solution  of  which  is  given  in  the  figure.  Spheri- 
cal waves  diverging  from  P  seem  to  diverge  from  the  point 
P  after  they  leave  the  lens.  So  P  is  the  virtual  image  of 
P.  The  image  of  the  object  0  P  is  in  this  position  virtual 
and  erect ;  it  is  also  magnified.  The  linear  magnification 


=  —  *  as  before ;    and  it  is   evident   that   if  Cf  =  Cf, 

OP  ^e 

i.  e.  if  Ce  —  o,  the  magnification  is  infinite. 

338.   e.  The  incident  waves  diverge  from  any  point  in  a 


339] 


REFRACTION 


449 


plane  perpendicular  to  the  axis  at  the  principal  focus. 
This  plane  is  called  the  "  focal  plane ; "  and,  if  the  point  0 
is  sending  out  spherical  waves,  they  will  emerge  from  the 
lens  as  plane  waves  with  their  wave-normal  parallel  to  the 
line  0  C,  joining  0  to  the  centre  of  the  lens.  This  becomes 
apparent  if  two  lines  of  disturbances  are  traced,  one  through 


FIG.  262. 

<7,  the  other  parallel  to  the  axis  and  then  refracted  through 
F\  for,  by  equal  triangles,  since  C  Ff  and  C  F  are  equal, 
the  two  emerging  lines  of  disturbances  are  parallel.  Con- 
versely, a  train  of  plane  waves  incident  upon  the  lens  will 
be  converged  to  a  focus  in  the  focal  plane  at  a  point  where 
a  line  through  the  centre  of  the  lens,  parallel  to  the  wave- 
normal,  meets  the  focal  plane. 

339.  /.  The  general  formula  may  also  be  expressed  in 
terms  of  distances.     For 


C  -  == 
1      00 


C  -    = 

~  O'C 


C  -  =  =  - 
OF 


"Hence 


/' 


where  u  is  positive,  if  the  object  is  on  the  same  side  of  the 
lens  as  the  surface  which  the  incident  waves  first  strike  ; 
v  is  positive  if  the  image  is  on  the  same  side  of  the  lens 
as  the  surface  from  which  the  waves  emerge  ;  /  is  essen- 
tially positive  for  ordinary  lenses.  Stated  in  other  words, 

15 


450 


THEORY   OF  PHYSICS 


[CH.  Ill 


u  is  positive  for  diverging  waves,  negative  for  converging ; 
v  is  positive  for  converging  waves,  negative  for  diverging. 

The  linear  magnification  obviously  equals  v  /  u. 

Combinations  of  Lenses.  Various  combinations  may  be 
arranged;  but  only  a  few  are  of  fundamental  importance. 

340.  1.  Microscopes.  A  microscope  is  an  instrument 
designed  to  magnify  the  apparent  size  of  objects.  The 
simplest  form  is  the  ordinary  "  magnifier,"  which  consists 
of  a  single  converging  lens  (or  converging  system),  which 
is  so  placed  that  the  object  comes  just  inside  the  focus. 
Then,  as  explained  in  Article  337,  a  magnified  image  of 
the  object  will  be  formed,  which  may  be  seen  by  the  eye 
of  an  observer. 

A  compound  microscope,  as  commonly  used  by  micro- 
scopists,  consists  of  two  converging  lenses  LI  and  Z2,  which 


+T+ 


FIG.  263. 

can  be  so  adjusted  that,  when  the  object  to  be  examined  is 
placed  outside  the  focus  of  Zi,  its  image  will  fall  just 
inside  the  focus  of  Z2.  Consequently,  there  will  be  a  real 
image  formed  by  Zx ;  and  this  will  be  magnified  by  Z?  into 
a  virtual  image.  The  first  lens  is  called  the  "  objective ; M 
the  second,  the  "  eye-piece."  In  the  figure  0  P  is  the  ob- 
ject ;  0'  F  is  the  real  image  formed  by  Zi ;  0"  F"  is  the 
virtual  magnified  image  formed  by  Z2. 


0"  P1 

The  linear  magnification  is    

OP 


But 


340] 


REFRACTION 


451 


o"  P"    o"  a 


O'P' 

therefore 


a  c* 


and 


0'  P'      0' 


and 


OP       OCi 


0"  P" 


x 


OP 


0'  C3  x  0 


In  actual  practice  0  Cl}  the  distance  from  the  object  to 
the  objective,  is  made  as  small  as  possible,  consistent  with 
its  being  beyond  the  focus ;  and  0"  C2  has  a  limiting  value 
depending  upon  the  distance  at  which  the  eye  of  an  ob- 
server can  see  the  image  clearest.  In  most  cases  this 
distance  is  about  fourteen  inches. 

The  instrument  must  be  so  designed  also  that  the  cone 
of  waves  produced  by  any  point  of  the  object  is  slightly 


FIG.  264. 

larger  than  the  aperture  of  the  pupil  of  the  eye,  when 
the  waves  finally  emerge  from  the  last  lens.  This  is  illus- 
trated in  the  figure.  The  size  of  the  objective  L\  is  always 
limited  by  some  diaphragm  or  stop ;  and  so  the  cone  of 
waves  which  start  from  0  will  have  the  path  as  shown  ; 
the  final  emerging  cone  of  waves  seeming  to  come  from 
0".  In  practice,  diaphragms  are  introduced  in  between 
the  two  lenses  so  as  to  cut  off  all  the  portions  of  the  lenses 
except  near  the  axis. 

It  should  be  noted  that  this  process  of  magnification 
cannot  continue  indefinitely ;  for  it  may  be  proved  that 
if  two  points  are  as  close  together  as  \  /  2,  half  the  wave- 


452 


THEORY  OF  PHYSICS 


[CH.  Ill 


length  of  the  light  used,  no  magnification  can  "  resolve  " 
them,  that  is,  show  them  as  two  distinct  points. 

341.  2.  Telescopes.  A  telescope  is  an  instrument  designed 
to  render  distant  objects  more  clearly  visible  to  the  eye. 
There  are  a  great  many  types  of  telescopes,  some  depending 
upon  the  use  of  concave  mirrors,  others  upon  lenses.  The 
former  are  called  "  reflectors  ; "  the  latter,  "  refractors."  Ee- 
fractors  are  of  two  types,  drawings  for  both  of  which  are 
given.  The  first  consists  of  two  converging  lenses  Zi  and 
Z2  so  placed  that  their  two  foci,  F'  and  /,  almost  coincide. 


F,  /  O' 


FIG.  265. 

The  waves  coming  from  a  distant  object  and  falling  upon 
Li  are  brought  to  a  focus  at  0'  P  just  beyond  the  focus  P. 
For  the  waves  converging  to  P  are  coming  from  a  point  P 
in  the  line  P  C\  so  far  away  that  the  waves  which  it  emits 
are  nearly  plane  when  they  reach  the  lens  ;  and,  similarly, 
0'  is  the  focus  of  nearly  plane  waves  emitted  by  a  source 
0,  very  far  away  on  the  line  0'  C\.  This  real  image  lies 
just  inside  the  focus  /  of  the  lens  Z2 ;  and  so  a  magnified 
virtual  image  will  be  formed.  The  power  of  the  telescope 
must  be  expressed  as  the  ratio  of  the  apparent  size  of  this 
virtual  image  0"  P"  to  the  apparent  size  of  the  original 
object  if  viewed  with  the  naked  eye  ;  and  these  sizes  may 
best  be  compared  by  means  of  the  angles  which  are  sub- 
tended at  the  eye  in  the  two  cases.  The  angle  subtended 
by  0"  P"  is  0"  Ci  P"  ;  that  subtended  by  the  distant  ob- 
ject is  0'  Ci  P'.  Therefore,  the  power  of  the  instrument  is 


453 


if  the  angles  are  small.     But  ft  0'  is  the 


342]  REFRACTION 

0"  C2  F'  ~CTO' 
0'C,P  "  ft  0' 
focal  length  of  Zi ;  and  ft  0'  is  the  focal  length  of  Z2  if  F' 
and /coincide  and  the  object  is  very  far  away.  If  these 
two  focal  lengths  are  FI  and  Fz,  the  power  of  the  telescope 
is  Fl  /F2.  The  first  lens  is  sometimes  called  the  "  object- 
glass  ; "  and  the  second,  the  "  eye-piece."  The  focal  length 
of  the  object-glass  must  be  large  so  as  to  have  great 
power ;  and  the  lens  itself  must  also  be  as  large  as  the 
nature  of  the  material  will  permit,  in  order  to  receive  as 
much  light  as  possible  from  the  distant  object. 

342.    The  second  type  of  refracting  telescope  consists  of 
two  lenses  Zi  and  Z2 ;  the  one  nearer  the  distant  object, 


FIG.  266. 

Zi,  is  converging ;  the  second,  Z2,  is  diverging.  They  are 
so  placed  that  the  two  foci  F1  and  ff  nearly  coincide,  as 
shown.  The  converging  lens  would  form  an  image  of  the 
distant  object  at  0'  P ;  but,  owing  to  the  presence  of  the 
diverging  lens,  the  waves  converging  toward  P'  are  so 
changed  as  to  become  diverging  from  a  virtual  image  P'. 
Similarly,  the  waves  converging  toward  0'  will  be  diverged 
by  Z2  until  they  seem  to  come  from  0" .  So  that  a  virtual, 
erect  image  0"  Pf  is  formed  of  the  distant  object.  This 
is  the  construction  and  principle  of  the  ordinary  "  opera- 
glass." 


454  THEORY  OF  PHYSICS  [CH.  Ill 

343.  3.  The  human  eye  is,  in  one  respect,  an  optical 
instrument,  because  it  consists  of  a  Combination  of  lenses 
which  focus  upon  the  retina  light-waves  coming  from  illu- 
minated objects.  The  eye  has  but  little  spherical  aberra- 
tion, owing  to  its  peculiar  shape  and  to  the  action  of  the 
iris,  which  takes  the  place  of  a  diaphragm ;  but  it  does 
have  considerable  chromatic  aberration.  Most  eyes  have 
power  of  "  accommodation,"  that  is,  of  altering  their  focal 
length  at  will  so  as  to  perceive  objects  at  different  distances 
away.  There  are,  however,  several  possible  optical  defects 
in  eyes,  which  may  arise  from  various  causes :  — 

a.  The  waves  may  be  brought  to  a  focus  in  front  of 
the  retina  instead  of  on  it.     Such  eyes  are  called  "  near- 
sighted,"  and   may   be  helped   by  the   use   of   diverging 
lenses. 

b.  The  waves  may  be  focused  back  of  the  retina.     Such 
eyes  are  called  "far-sighted,"  and  may  be  helped  by  the 
use  of  converging  lenses. 

c.  The  focus  may  be  different  for  different  sections  of 
the  eye.     For  instance,  if  the  dial  of  a  clock  is  looked 
at,  an  eye  may  see  the  figures  2  and  8  clearly,  but  may 
not   see   the   5   and   11   sharply.     Such   eyes   are   called 
"  astigmatic,"  and  may  be  helped  by  the  use  of  cylindrical 
lenses. 

d.  In  normal  eyes,  the  images  formed  by  the  two  eyes 
of  the  same  object  fall  upon  two  points  in  the  retinas 
which  are  said  to  "  correspond  ; "  and  only  one  visual  im- 
pression is  made.     But  it  may  happen,  owing  to  muscular 
troubles,  that  the  images  formed  by  the  two  eyes  do  not 
correspond ;  and  so  two  visual   impressions   are   seen   of 
the  same  object.     Such  eyes  may  be  helped  by  the  use 
of  prisms. 


CHAPTEE   IV 
DISPERSION  —  SPECTRA 

344.  Dispersion.  It  has  been  shown  in  the  last  chapter 
that,  when  a  train  of  plane  waves  passes  through  a  prism, 
the  direction  of  the  wave-normal  is  changed ;  that  is,  there 
is  "  deviation."  This  deviation  was  shown  to  depend  upon 
several  conditions,  —  the  wave-number  of  the  incident 
waves,  the  angle  of  incidence,  the  material  of  the  prism, 
and  the  angle  of  the  prism.  It  was  further  proved  that 
waves  of  greater  wave-number  (i.  e.  of  less  wave-length) 
are  (in  general)  deviated  more  than  those  of  a  smaller 
wave-number.  If,  then,  two  trains  of  waves  of  different 
wave-number  fall  upon  the  same  prism  at  the  same  angle 
of  incidence,  they  will  be  deviated  differently;  and  the  dif- 
ference in  the  angles  of  deviation,  that  is,  the  angle  between 
the  normals  of  the  two  emerging  waves,  is  called  their  "  dis- 
persion." As  might  be  expected,  the  dispersion  of  the  same 
two  trains  of  waves  is  different  for  different  angles  of  inci- 
dence and  for  different  prisms  ;  and,  for  the  same  angle  of 
incidence  and  the  same  prism,  the  dispersion  is  different 
for  different  trains  of  waves,  even  if  they  have  the  same 
difference  in  their  wave-numbers.  Since,  when  two  trains 
of  plane-waves  in  the  same  direction  but  of  different  wave- 
numbers  are  refracted  through  a  prism,  they  emerge  in 
different  directions,  it  follows  that,  if  these  same  two  trains 
of  waves  are  made  to  fall  upon  the  prism  at  the  same  angles 
as  those  at  which  they  emerged,  they  will  be  refracted  out 
so  as  to  emerge  with  their  wave-normals  in  the  same 
direction. 


456  THEORY  OF  PHYSICS  [CH.  IV 

345.  Pure  Spectrum.  If  waves  coming  from  a  source 
which  appears  white  to  our  eyes  are  made  to  pass  through 
a  prism  and  then  fall  on  a  screen,  it  is  observed  that  the 
illumination  on  the  screen  is  not  white,  but  colored.  If 
the  source  of  light  is  a  point,  there  will  be,  as  explained  in 
Article  325,  a  series  of  virtual  images  corresponding  to 
each  train  of  waves  of  a  definite  wave-number,  which  is 
emitted  by  the  source ;  and  each  one  of  these  images  will 
be  the  centre  of  a  train  of  diverging  waves,  which  will  illu- 
minate the  screen.  There  will,  therefore,  be  an  overlap- 


FIG.  267. 

ping  of  the  effects  due  to  the  separate  waves.  This  may 
be  prevented  by  interposing  between  the  virtual  images 
and  the  screen  a  converging  lens,  which  will  focus  on  the 
screen  the  waves  diverging  from  the  images.  There  will 
be,  under  these  conditions,  a  separation  of  the  waves,  so 
that,  corresponding  to  each  virtual  image  of  the  source, 
there  is  a  definite  point  of  illumination  on  the  screen.  All 
the  different  trains  of  waves  emitted  by  the  source  will 
thus  be  separated  and  spread  out,  as  it  were,  on  the  screen, 
each  illuminated  point  receiving  only  waves  of  one  wave- 
number.  In  practice,  the  source  of  light  is  not  a  point, 
but  is  a  narrow  slit  placed  parallel  to  the  edge  of  the 
prism,  and  in  front  of  the  flame  or  light ;  and  the  images 


346]  DISPERSION  —  SPECTRA  457 

on  the  screen  are,  then,  not  points  but  narrow  lines  par- 
allel to  the .  slit.  When  the  waves  from  any  source  are 
thus  resolved  into  their  components  by  a  slit,  a  prism  and 
a  lens,  the  resulting  single  waves  are  said  to  form  the 
"  spectrum  "  of  the  source.  Tf  the  slit  is  extremely  narrow, 
its  images  on  the  screen  will  be  narrow ;  and  there  will 
not  be  much  overlapping  of  the  separate  images  due  to 
the  separate  trains  of  waves.  Such  a  spectrum  is  called 
"  pure "  to  distinguish  it  from  one  where  there  is  over- 
lapping, produced  either  by  width  of  slit  or  absence  of 
lens.  The  purest  spectrum  is  obtained  when  the  incident 
waves  fall  upon  the  prism  as  nearly  as  possible  at  the 
angle  which  corresponds  to  minimum  deviation  ;  for,  under 
these  conditions,  the  virtual  images  of  the  slit  are  the 
sharpest  possible.  (See  the  end  of  Art.  325.) 

If  the  source  is  one  of  ordinary  white  light,  such  as  a 
white-hot  solid,  and 'if  the  prism  is  one  of  ordinary  glass,  its 
spectrum,  as  shown  on  the  screen,  wTill  be  a  series  of  waves 
which  produce  in  our  eyes  the  sensations  blue,  green,  yel- 
low, and  red,  as  well  as  all  intermediate  shades ;  and  so 
"  white  light "  may  be  regarded  as  a  mixture  of  all  those 
waves  which  produce  these  individual  sensations.  Those 
waves  which  produce  the  sensation  blue  will,  as  noted 
several  times,  be  deviated  more  than  those  which  produce 
the  sensations  green  and  red,  if  the  prism  is  any  ordi- 
nary one;  and  the  arrangement  of  the  colors  in  the  spec- 
trum will  be  in  the  order  of  the  wave-numbers  of  the 
corresponding  waves.  In  the  spectrum  of  the  white-hot 
•solid  there  are,  of  course,  other  waves  than  those  which 
appeal  to  our  sense  of  vision ;  some  are  longer  than  the 
"  visible  waves,"  and  may  be  detected  by  their  heating 
effect,  —  they  are  called  the  "  ultra-red  waves ;  "  others  are 
shorter  than  the  visible  waves,  and  may  be  detected  by 
their  action  on  certain  chemicals  such  as  a  photographic 
plate,  —  they  are  called  the  "  ultra-violet  waves." 

346.    Anomalous   Dispersion.      There   are   certain    prisms 

15* 


458  THEORY  OF  PHYSICS  [CH.  IT 

which  do  not  disperse  the  waves  in  the  order  of  their 
wave-numbers;  and  such  dispersion  is  called ." anomalous.'" 
If  waves  from  a  white-hot  solid  pass  through  such  a 
prism,  there  is  always  a  gap  in  the  spectrum,  perhaps 
several,  showing  that  certain  waves  have  riot  come  through 
but  have  been  "  absorbed "  by  the  prism ;  and  the  waves 
on  each  side  of  a  gap  are  displaced,  so  that  those  waves 
which  are  longer  than  the  ones  absorbed  are  deviated  more 
than  they  ordinarily  would  be,  and  those  which  are  shorter 
are  deviated  less.  As  seen  on  a  screen,  or  as  viewed  by  the 
naked  eye,  the  colors  on  each  side  of  the  absorption  band 
are  shifted  towards  and  over  each  other.  There  is  a  most 
intimate  connection  between  this  phenomenon  and  certain 
others  to  be  described  later.  (See  Art.  355.) 

An  illustration  of  anomalous  dispersion  is  afforded  by  a 
prism  of  alcoholic  solution  of  fuchsine  (aniline-red),  The 
greenish-blue  is  absorbed ;  and  the  order  of  the  colors, 
according  to  deviation,  is  violet,  blue,  red,  yellow. 

347.  Achromatism.  As  has  been  repeatedly  mentioned,  a 
prism  or  a  lens  has  chromatic  aberration ;  that  is,  the  image 
of  any  source  of  light  as  formed  by  it  depends  upon  the 
wave-number  of  the  waves ;  and,  if  a  source  of  white  light 
is  used,  there  will  be  a  series  of  colored  images.  This  is,  of 
course,  most  inconvenient  for  many  purposes ;  for  instance 
in  the  method  for  securing  a  pure  spectrum,  an  ordinary 
lens  would  not  focus  all  the  waves  at  the  same  dis- 
tance from  the  lens.  It  is.  possible,  however,  to  make 
combinations  of  lenses  which  will  largely  obviate  this 
difficulty. 

As  stated  in  Article  344,  it  may  be  proved  by  experi- 
ment that  the  dispersion  of  any  two  trains  of  waves  de- 
pends upon  the  prism,  its  material  and  angle,  and  upon  the 
angle  of  incidence.  Therefore  it  may  be  possible  to  make 
two  prisms  of  different  material  and  angle,  such  that  they 
will  produce  the  same  dispersion  of  these  two  trains  of 
plane  waves  ;  although,  to  produce  this,  the  angle  of  inci- 


347]  DISPERSION  —  SPECTRA  459 

dence  upon  the  two  prisms  must,  in  general,  be  different, 
and  the  angles  of  emergence  will  be  different.  If  these 
two  trains  of  plane  waves  are  incident  together  upon  one 
prism,  they  will  emerge  dispersed  through  a  certain  angle ; 
and,  if  they  now  fall  upon  the  other  prism  which  is  so 
placed  as  to  refract  them  together,  they  will  emerge  from 
it  with  their  wave-normals  parallel,  as  shown ;  for  one 
prism  simply  neutralizes  the  dispersion  of  the  other.  The 


FIG.  268. 

direction  of  the  wave-normal  of  the  emerging  waves  is  not, 
in  general,  the  same  as  that  of  the  incident  waves ;  and, 
therefore,  there  is  deviation  but  no  dispersion.  By  a  suit- 
able choice  of  prisms,  it  is  possible,  of  course,  to  prevent 
the  dispersion  of  two  trains  of  waves  of  any  wave-num- 
bers ;  but,  if  in  the  incident  waves  there  are  some  of  other 
wave-numbers,  all  these  others  will  be  dispersed  slightly, 
although  not  so  much  as  ordinarily  with  a  single  prism. 
Thus,  if  waves  from  a  "  white-hot "  body  are  falling  on 
the  pair  of  prisms,  two  trains  of  waves  will  emerge  in 
the  same  direction ;  and  the  others,  in  slightly  different 
directions. 

In  a  perfectly  similar  manner  it  is  possible  to  combine 
two  lenses  of  different  material  and  curvatures,  one  con- 
verging, the  other  diverging,  so  that  the  combination  brings 
to  the  same  focus  any  two  trains  of  waves.  This  is  pos- 
sible because,  as  shown  for  the  two  prisms,  there  is,  in 
general,  deviation  produced  by  the  combination  ;  and,  there- 
fore, so  far  as  the  ordinary  laws  of  prisms  and  lenses  are 
concerned,  the  combination  of  the  two  produce  the  same 


460  THEORY  OF  PHYSICS  [CH.  IV 

effect  as  one  if  suitably  chosen.  Consequently,  plane 
waves  will  be  brought  to  a  definite  focus,  etc.  Such  a 
combination  is  called  an  "  achromatic  lens,"  and  is  said  to 
be  "  corrected  for  "  those  two  definite  wave-numbers.  The 
other  waves  will  not  be  brought  to  the  same  focus,  but 
will  not  deviate  far  from  it.  If  a  lens  is  to  be  used  for 
visual  work,  it  is  generally  corrected  for  the  two  strongest 
colors  in  the  visible  spectrum  which  are  some  distance 
apart,  perhaps  the  blue  and  yellow.  Then  these  corre- 
sponding waves  will  be  brought  to  the  same  focus,  and  the 
other  waves  will  come  nearly  to  the  same  point ;  so  that, 
if  waves  from  a  white-hot  body  pass  through  such  a  lens, 
the  light  at  the  focus  will  be  nearly  white,  although  it 
still  is  slightly  colored.  These  remaining  colors  are  some- 
times called  "  secondary  colors."  A  lens  which  is  to  be 
used  for  photographic  work  is,  naturally,  corrected  for  two 
waves  which  produce  great  photographic  action. 

348.  Direct- Vision  Spectroscope.  Two  prisms  of  different 
material  and  angle  may,  of  course,  produce  the  same 
deviation  for  any  one  train  of  waves ;  but,  since  their  dis- 
persion will  be  different,  the  deviation  of  any  other  train 
of  waves  will  not  be  the  same  in  both  prisons.  Therefore, 
if  these  two  prisms  are  combined  so  that  one  tends  to 
neutralize  the  deviation  produced  by  the  other,  that  is,  if 
they  are  placed  each  with  its  edge  near  the  base  of  the 
other,  they  will  form  a  system  such  that,  if  the  waves 
from  a  white-hot  body  pass  through  them,  one  particular 
train  of  waves  may  not  be  deviated  at  all,  but  will  emerge 
parallel  to  its  original  direction,  while  the  others  will  be 
slightly  deviated  from  this  direction.  Consequently  there 
will  be  dispersion.  If  several  prisms  are  used,  consider- 
able dispersion  may  be  secured ;  and  by  suitable  arrange- 
ments of  a  slit  and  lens  a  pure  spectrum  of  the  source 
will  be  obtained.  Such  an  instrument  is  called  a  "  direct- 
vision  spectroscope,"  because  the  entire  apparatus  for  view- 
ing the  spectrum  is  all  arranged  in  one  straight  line. 


349]  DISPERSION  —  SPECTRA  461 

349.  Ordinary  Spectroscope.  The  ordinary  spectroscope,  or 
instrument  for  the  study  of  the  spectra  emitted  by  differ- 
ent sources,  is  a  simple  modification  of  the  apparatus  de- 
scribed before,  in  Article  345,  for  producing  a  pure  spectrum. 
It  consists  of  three  separate  parts :  the  collimator,  the 
prism  (or  train  of  prisms),  and  the  telescope.  The  colli- 
mator is  a  metal  tube  at  one  end  of  which  is  placed  the 
slit,  and  at  the  other  an  achromatic  converging  lens,  the 
tube  being  of  such  a  length  that  the  slit  is  at  the  prin- 
cipal focus  of  the  lens.  It  is  carried  by  a  rigid  arm 
which  can  turn  around  an  axis  perpendicular  to  the 


FIG.  269. 


platform  which  carries  the  prisms.  This  axis  is  called 
the  "  axis  of  the  instrument ;  "  and  the  collimator  is  ad- 
justed perpendicular  to  it.  Several  prisms  are  generally 
used,  so  as  to  produce  greater  dispersion ;  and  the  different 
prisms  are  all  placed  on  a  platform  with  their  edges 
parallel  to  the  axis  of  the  instrument,  and  on  the  same 
level  as  the  collimator.  They  are,  further,  all  joined  by 
such  a  link-work  that,  however  the  collimator  is  turned 
around  the  axis  of  the  instrument,  the  prisms  are  so 
moved  that  waves  from  the  collimator  will  fall  upon  the 
first  prism  and  upon  the  others  in  turn,  at  the  angle  which 
corresponds  to  minimum  deviation.  The  telescope  is  an 
ordinary  instrument  with  two  converging  lenses,  as  ex- 
plained in  Article  341.  It  is  carried  by  a  rigid  arm  which 
is  movable  about  the  axis  of  the  instrument ;  and  it  is 
adjusted  perpendicular  to  this  axis,  and  at  the  same  level 
as  the  collimator. 


462  THEORY  OF  PHYSICS  [CH.  IV 

If,  now,  the  slit  is  made  very  narrow,  and  is  illuminated 
by  any  source,  the  corresponding  waves  will  leave  the  lens 
of  the  collimator  with  plane  wave-fronts,  will  fall  upon 
the  prisms  at  the  angle  which  corresponds  to  minimum 
deviation,  will  leave  the  prisms  as  if  coming  from  a  series 
of  virtual  images  at  an  infinite  distance,  and  will  be  fo- 
cused by  the  object-glass  of  the  telescope  into  a  series  of 
lines  in  the  focal  plane,  each  line  corresponding  to  a  train 
of  waves  of  a  definite  wave-number.  This  series  of  images 
is  then  magnified  by  the  eye-piece  of  the  telescope.  In 
this  manner  the  nature  of  any  source  of  light  may  be 
studied. 

It  is  convenient,  of  course,  to  have  some  means  of  lo- 
cating or  distinguishing  the  different  trains  of  waves  and 
their  images ;  and  one  of  two  methods  is  commonly  adopted  : 
one  is  to  throw  into  the  focal  plane  of  the  telescope  a  fixed 
scale  divided  into  convenient  lengths  ;  the  other  is  to  illu- 
minate part  of  the  slit  with  some  standard  source,  and  to 
compare  with  its  spectrum  the  waves  coming  through  the 
other  part  of  the  slit,  due  to  the  source  to  be  investigated. 

350.  Continuous  and  Discontinuous  Spectra.  When  the  waves 
coming  from  various  sources  are  thus  analyzed  by  means 
of  a  spectroscope,  it  is^observed  that  there  are  two  classes  of 
spectra,  —  one  where  all  possible  waves  seem  to  be  present, 
the  oth^r  where  there  are  certain  gaps  due  to  the  absence 
of  particular  waves.  The  former  is  called  a  "  continuous  " 
spectrum ;  and  it  is  proved  by  experiment  that  a  solid 
body  emits  such  a  one.  That  is,  if  a  solid  is  white-hot,  it 
emits  all  the  waves  which  lie  in  the  visible  spectrum,  as 
well  as  others,  both  longer  and  shorter,  there  being  no 
waves  absent  between  the  limiting  wave-lengths.  The 
other  kind  of  a  spectrum  is  called  "  discontinuous  ; "  and 
it  is  emitted  by  vapors  and  gases,  when  they  are  excited 
to  luminescence  in  any  way,  e.  g.  by  passing  an  electric 
spark  through  them. 

This  difference  between  solids  and  gases  is  what  might 


351]  DISPERSION  —  SPECTRA  463 

be  expected  on  the  kinetic  theory  of  matter.  In  gases  the 
molecules  have  certain  "  free  paths  ; "  and  while  the  mole- 
cules are  moving  along  them,  their  parts  may  vibrate  freely 
in  definite  periods,  thus  producing  trains  of  waves  of  defi- 
nite wave-numbers.  In  a  solid,  however,  the  molecules 
are  so  close  together,  and  so  connected,  that  any  definite 
periodic  vibration  is  impossible,  and  all  possible  periods 
and  corresponding  waves  occur. 

The  spectrum  of  any  gas  or  vapor  is  perfectly  charac- 
teristic of  itself ;  it  consists  of  many  isolated  waves  which 
are  peculiar  to  itself.  So,  by  examining  the  spectrum  of 
.any  gaseous  source,  it  is  often  possible  to  detect  what  gases 
or  vapors  are  present  there. 

351.  Emission  and  Absorption  Spectra.  The  waves  which 
are  emitted  by  any  source,  such  as  an  incandescent  gas, 
are  said  to  give  an  "  emission  "  spectrum  when  analyzed  by 
a  spectroscope. 

If  waves  from  a  white-hot  solid  are  made  to  enter  any 
substance,  the  emerging  waves  may  be  different  from  those 
entering ;  some  waves  may  have  been  "  absorbed."  This 
may  be  tested  by  analyzing  the  emerging  waves  by  means 
of  a  spectroscope.  The  white-hot  solid  emits  a  con- 
tinuous spectrum ;  and,  consequently,  any  gaps  in  the 
spectrum  must  be  due  to  the  absorption  of  the  correspond- 
ing waves  by  the  substance  interposed  between  the  slit  of 
the  spectroscope  and  the  white-hot  source.  These  waves 
which  are  absorbed  by  any  substance  are  called  the  "  ab- 
sorption spectrum"  of  that  substance.  In  general,  the 
absorption  spectrum  of  a  liquid  consists  of  whole  groups  of 
waves  ;  and  so  there  are  wide  gaps  in  the  spectrum  as  seen 
in  the  spectroscope.  But  the  absorption  spectrum  of  a 
vapor  consists  of  separate  isolated  waves. 

There  is  a  most  interesting  connection  between  the 
emission  spectrum  and  absorption  spectrum  of  the  same 
vapor  at  the  same  temperature  ;  in  fact,  they  are  identical. 
That  is,  those  waves  which  a  vapor  emits  are  exactly  those 


464  THEORY  OF  PHYSICS  [CH.  IV 

which  it  will  absorb  out  of  all  the  waves  of  a  continuous 
spectrum.  This  is  but  a  special  application  of  the  general 
law  of  radiation  and  absorption,  which  was  explained  in 
HEAT  (Art  206). 

The  intensity  of  the  emission  or  the  absorption  depends, 
of  course,  upon  the  temperature  of  the  vapor ;  the  higher 
the  temperature,  so  much  the  more  intense  is  the  emission. 

352.  Fraunhofer's  Lines.  The  spectrum  of  the  waves 
coming  from  the  sun  is  found  on  examination  to  be  an 
absorption  one ;  and  the  explanation  is  obvious :  the  sun 
consists  of  a  white-hot  central  mass  which  emits  a  con- 
tinuous spectrum,  and  which  is  surrounded  by  layers  of 
absorbing  vapors.  These  vapors,  of  course,  emit  definite 
waves  ;  but,  as  their  temperature  is  so  much  lower  than 
that  of  the  central  mass,  those  waves  which  come  through 
from  the  inside,  that  is,  those  which  are  unabsorbed,  are  so 
intense,  compared  with  those  emitted  by  the  outer  layers* 
that  the  latter  produce  no  effect  comparable  with  the 
former.  When  sunlight  is  analyzed,  then,  by  means  of  a 
spectroscope,  there  will  be  in  the  spectrum  dark  lines, 
which  are  parallel  to  the  slit,  and  which  mark  the  ab- 
sorbed waves.  These  lines  are  called  "Fraunhofer's  Lines," 
from  the  name  of  the  German  astronomer  who  first  care- 
fully studied  them. 

Since  the  waves  absorbed  by  any  vapor  are  exactly  those 
which  it  emits,  it  is  seen  that,  if  it  can  be  proved  that 
certain  vapors  emit  waves  whose  absorption  lines  are  in 
the  solar  spectrum,  that  fact  demonstrates  the  existence  of 
those  vapors  in  the  layers  around  the  central  mass  of  the 
sun.  The  emission  spectra  of  a  great  many  vapors  have 
been  carefully  examined,  and  compared  with  the  solar 
spectrum;  and  almost  all  the  Fraunhofer  lines  have  been 
accounted  for.  The  following  substances  undoubtedly  ex- 
ist on  the  sun :  calcium,  iron,  hydrogen,  sodium,  nickel, 
magnesium,  cobalt,  silicon,  aluminium,  titanium,  chromium, 
manganese,  carbon,  and  perhaps  thirty  others.  Antimony, 


lit 


355]  DISRERSION  — SPECTRA  465 

arsenic,  bismuth,  gold,  boron,  mercury,  and  some  others  do 
not  occur  so  far  as  is  known. 

In  a  similar  manner  the  nature  of  the  substances  on 
many  of  the  stars  nas  been  studied ;  for  in  some  cases 
their  spectra  are  emission,  in  others  absorption. 

353.  Absorption.     When  waves  from  a  white-hot  solid 
fall  upon  any  substance,  it  may  happen  that  some  waves 
of    particular   wave-numbers    do   not   enter   the   surface ; 
others  that  do  enter  may  not  pass  through  to  the  further 
side ;  and,  consequently,  of  those  waves  which  are  thus 
absent  from  the  emerging  trains,  some  are  "  absorbed  "  at 
the  surface,  others  in  the  interior  of  the  substance.     (This 
surface-effect  has  no  connection  with  ordinary  reflection, 
for  in  that  all  waves  are  reflected  alike.) 

Absorption  implies  the  removal  from  the  entering  waves 
of  some  particular  trains ;  and,  as  waves  carry  energy,  an 
explanation  must  be  given  of  the  way  in  which  this  ab- 
sorbed energy  becomes  manifest. 

354.  a.  Surface-absorption.     An  illustration  of  this  is 
given  by  any  metallic  surface,  or  by  a  surface  of  almost 
any  of  the  aniline  dyes.     If  waves  from  a  source  of  white 
light  fall  upon  such  a  surface,  certain  definite  waves  are 
reflected  at  once,  apart  from  a  certain  percentage  of  all  the 
waves  which  is  reflected  owing  to  ordinary  reflection.    Thus 
a  surface  of  gold  reflects  those  waves  which  produce  in  our 
eyes  the  sensation  yellow.     The  other  waves  enter  the 
gold  and  are  soon  absorbed  there,  unless  the  gold  is  in  the 

*  form  of  a  film. 

If  a  sufficiently  thin  film  of  a  metal  is  made,  it  is  trans- 
parent to  certain  waves ;  and  it  is  possible  also  to  make 
prisms  of  most  metals.  It  is  observed  that  such  prisms 
have  "  anomalous  dispersion "  (see  Art.  346).  A  prism 
made  of  a  solution  of  an  aniline  dye  also  has  anomalous 
dispersion. 

355.  &.   Ordinary  absorption.     As  trains  of  waves  pass 
through  any  medium,  certain  waves  are  as  a  rule  absorbed ; 


466  THEORY  OF  PHYSICS  [CH.  IV 

and  their  energy  may  be  spent  in  several  ways.  In  gen- 
eral, it  is  used  in  producing  "  heat-effects  "  in  the  absorbing 
medium.  Other  waves  are  not  absorbed,  and  can  pass 
through  the  medium  freely. 

Thus,  if  waves  from  a  source  of  white  light  enter  a  leaf 
of  a  tree,  certain  waves  are  absorbed,  and  others  pass 
through,  or  are  reflected  out  from  the  interior  if  they 
strike  any  reflecting  bodies  inside.  The  waves  which 
emerge  produce  color  sensations  in  the  eye  of  an  observer; 
green,  in  the  case  of  a  leaf.  This  is  the  explanation  of  the 
colors  of  most  natural  objects. 

The  amount  of  the  absorption  of  any  particular  train  of 
waves  depends  upon  the  distance  through  which  the  waves 
pass;  and  so  it  often  happens  that  thin  and  thick  layers  of 
the  same  substance  have  different  colors,  if  it  absorbs  the 
two  corresponding  trains  of  waves  differently. 

356.  c.  Fluorescence.      In    certain    substances   (perhaps 
all),  some  of  the  energy  received  by  the  absorption  of  the 
waves  is  not  spent  in  heat-effects,  but  in  emitting  new 
waves.     Such  substances  are  called  "fluorescent."     It  is  a 
general  law  that  the  wave-length  of  the  waves  absorbed  is 
always  shorter  than  that  of  the  waves  emitted  by  means  of 
the  corresponding  energy.     Thus  "canary  glass,"  a  glass 
containing  uranium  oxide,   absorbs   the   blue  waves    and 
emits  greenish-yellow. 

357.  d.  Phosphorescence.     Most  fluorescent  bodies  emit 
the  fluorescent  waves  only  while  they  are  absorbing  other 
waves ;  but  certain  of  them  continue  to  emit  waves  for 
some  time  after  absorption  has  ceased.     Such  bodies  are 
called   "phosphorescent."      They   store   up   the   absorbed 
energy,  and  emit  it  slowly.     Many  sulphides  are  phospho- 
rescent ;  and,  if  they  are  exposed  to  sunlight  for  a  few  min- 
utes, and  are  then  carried  into  a  dark  room,  they  will 
continue  to  shine  for  many  hours. 

358.  e.  Reflection  "by  Fine  Particles.     If  a  cloud  of  fine 
particles  is  interposed  in  a  beam  of  white  light,  it  may  be 


358]  DISPERSION  —  SPECTRA  467 

observed  that,  when  looked  at  from  the  side,  the  cloud  ap- 
pears colored  blue.  This  is  due  to  the  reflection  and  scat- 
tering by  the  fine  particles  of  the  shortest  waves,  those 
which  produce  in  the  eye  the  sensation  blue.  The  other 
waves  pass  by  or  around  the  particles ;  for  a  body  can 
reflect  only  those  waves  which  are  considerably  smaller 
than  its  own  dimensions.  This  is  the  explanation  of  the 
color  of  the  blue  sky. 


CHAPTEE  V 

COLORS  —  COLOR-SENSATION 

359.  Complementary  Colors.  It  has  been  shown  that, 
when  the  waves  from  a  white-hot  solid  are  analyzed  by  a 
spectroscope,  there  is  a  continuous  spectrum,  containing 
waves  of  all  possible  wave-numbers  between  certain  limits. 
Corresponding  to  any  one  of  these  wave-numbers  which  is 
in  the  visible  spectrum,  there  is  a  definite  sensation  of 
color.  So  white  light  is  commonly  said  to  be  a  mixture 
of  a  great  many  colors.  If  one  of  these  trains  of  waves 
is  absent  from  the  white  light,  the  other  waves  will 
combine  to  produce  in  the  eye  of  an  observer  a  certain 
color-sensation,  which  is  called  the  "  complementary " 
color  to  that  which  the  absent  train  would  produce. 
Thus,  red  and  greenish-blue  are  complementary  colors  \ 
so  are  yellow  and  blue. 

Conversely,  if  these  two  sensations  are  produced  simul- 
taneously iii  the  eye,  it  will  receive  the  sensation  of  white 
light.  (This  has  nothing  at  all  to  do  with  the  color-effect 
of  mixing  paints  or  colors ;  for  in  that  there  is  simply 
double  absorption  of  the  waves,  and  the  color-sensation 
produced  corresponds  to  those  waves  which  are  not  ab- 
sorbed by  either  paint.) 

In  a  similar  manner,  if  the  waves  which  remain  after 
one  train  is  removed  are  separated  into  two  sections,  each 
section  will  produce  a  definite  color-sensation  ;  and  white 
light  may  be  said  to  be  resolved  into  three  colors.  It  may 
be  proved  by  experiment  that,  using  different  intensities 


360]  COLORS  — COLOR-SENSATION  469 

of  any  three  colors  which  together  compose  white  light,  it 
is  possible  to  produce  any  color-sensation  desired. 

The  simplest  method  of  combining  color-sensations  is  to 
paint  sectors  of  a  circular  disc  with  the  different  colors, 
and  then  rapidly  to  revolve  the  disc  in  its  own  plane. 
Under  these  conditions  the  eye  receives  practically  simul- 
taneous sensations  from  all  the  colors. 

360.  Colors  of  Objects.  The  colors .  of  objects  depend 
upon  three  things,  —  the  nature  of  the  light  in  which  they 
are  viewed,  the  absorption  of  the  object,  the  peculiarities 
of  the  eye  of  the  observer. 

An  object  is  ordinarily  called  white  if  it  reflects  all  the 
visible  waves ;  and  so,  if  such  an  object,  like  a  piece  of 
paper,  receives  only  waves  which  produce  the  sensation 
yellow,  the  object  will  appear  yellow.  The  names  of  the 
colors  associated  with  all  natural  objects  depend  upon 
their  illumination  by  white  light.  A  stick  painted  red 
will  appear  black  if  placed  in  any  colored  light  except  red, 
unless  in  producing  this  red-colored  paint  some  other  color 
has  been  mixed  in ;  in  which  case,  if  placed  in  light  of 
that  color,  the  stick  will  appear  to  be  of  the  same  color. 

There  are  as  many  kinds  of  absorption-colors  as  there 
are  modes  of  absorption.  A  film  of  gold-leaf  appears  yellow 
if  white  light  is  reflected  from  it,  owing  to  surface-absorp- 
tion ;  and  the  complementary  waves  will  pass  through  so 
that  the  transmitted  light  has  the  color  blue.  As  previ- 
ously explained,  the  colors  of  almost  all  natural  objects 
depend  simply  upon  the  withdrawal  of  one  or  more  colors 
by  absorption  in  the  interior  of  a  body.  In  certain  cases, 
moreover,  the  effect  is  complicated  by  the  presence  of 
fluorescent  colors. 

The  normal  eye  is  able  to  distinguish  a  continuous 
gradation  of  colors  from  the  darkest  violet  to  the  deepest 
Ted,  and  to  distinguish  some  colors  which  do  not  appear 
in  the  spectra  of  bodies,  such  as  purple  and  brown.  But 
there  are  people  so  unfortunate  as  not  to  be  able  to 


470  THEORY   OF   PHYSICS  [CH.  V 

distinguish  all  these  colors ;  and  such  are  called  "  color- 
blind." It  happens,  not  unfrequently,  that  people  cannot 
distinguish  red  from  green ;  and  therefore,  in  the  case  of 
all  colors,  such  people  would  have  peculiar  and  individual 
sensations. 

361.  Perception  of  Color.  In  a  normal  eye  it  is  possible 
to  distinguish  two  entirely  different  sensations,  —  one  of 
gray,  and  one  of  color.  Any  train  of  waves  which,  if  in- 
tense, would  produce  a  color-sensation,  will,  if  very  weak, 
produce  the  gray  sensation  ;  and  again  there  are  portions 
of  the  retina  of  the  eye  where  only  gray  can  be  perceived. 

It  is  now  a  recognized  fact  that  the  sensation  of  color  is  a 
differentiation  of  the  gray  sensation  ;  and  that,  in  all  proba- 
bility, different  structures  of  the  retina  have  particular  func- 
tions to  perform  in  these  perceptions.  It  seems  probable 
that  the  so-called  "cones"  of  the  retina  are  the  instru- 
ments of  color-sensation  ;  and  that  the  "  rods  "  are  for  the 
gray  sensation. 


CHAPTER  VI 

INTERFERENCE  —  DIFFRACTION 

362.  Interference  Due  to  Two  Sources.  In  Young's  inter- 
ference experiment,  as  described  in  Chapter  L,  Article  300, 
two  slits  in  a  screen  were  illuminated  by  light  from  a  par- 
allel slit  in  another  screen,  and  the  slits  were  so  arranged 
that  the  two  slits  received  identically  the  same  illumina- 
tion. Under  these  conditions  there  were  dark  and  bright 
bands  on  the  screen,  if  the  light  was  homogeneous,  that  is, 
if  the  train  of  waves  had  a  definite  wave-number.  It  was 
absolutely  essential  that  the  two  slits  should  have  the 
same  illumination  in  every  way ;  that  is,  if  there  is  any 
change  in  one,  there  is  the  same  change  in  the  other,  etc. 
For,  if  two  different  sources  of  light  are  used,  e.  g.  if  two 
candle  flames  are  placed  behind  the  two  slits,  there  will 
no  longer  be  interference.  Any  source  of  light  emits  the 
waves  irregularly ;  and,  if  there  is  to  be  complete  interfer- 
ence between  waves  coming  from  two  sources,  it  is  obvious 
that  the  two  sources  must  undergo  identically 
the  same  changes. 

To  secure  interference,  then,  between  waves 
from  two  sources,  these  two  must  be  identical. 
Various  methods  have  been  devised  to  satisfy 
this  essential  condition. 

a.  The  simplest  apparatus  is,  as  already  de- 
scribed, a  slit  on  one  screen,  and  two  parallel 
slits  in  a  second  screen  parallel  to  the  first ;  the 
second  screen  being  so  placed  that  the  opaque  portion  be- 
tween the  two  slits  is  exactly  opposite  the  first  slit.     If  a 


B 


472 


THEORY  OF  PHYSICS 


[CH.  VI 


source  of  homogeneous  waves  is  placed  in  front  of  the  first 

slit,  the  two  slits  will  receive  identical  illumination,  and 

so  be  identical  sources. 

I.  A  slit  is  placed  parallel  to  the  edge  of  a  "  biprism."    A 

biprism,  of  which  a  section  is  shown,  consists,  as  it  were, 
of  two  very  thin  prisms,  exactly  alike, 
joined  at  their  bases.  The  slit  is  placed 
opposite  the  thickest  part  of  the  bi- 
prism ;  and  waves  from  any  source  in 
front  the  slit  will  fall  upon  the  prism, 
and  will  emerge  as  if  coming  from 
two  virtual  slits  at  A  and  £,  as  shown. 
These  two  virtual  images  are  identical ; 
and  so  are  the  trains  of  waves  which 


B 


FIG.  271. 


apparently  come  from  them. 

c.  Two    plane    mirrors    are    in- 
clined  to   each    other   at   a   most 
obtuse  angle,  and  a  slit  is  placed 
so   as   to   illuminate  both.     There 
will  be  an  image  of  the  slit  in  each 
mirror ;  and  the  two  images,  A  and 
B,  will  be  identical.     The  adjust- 
ments for  this  experiment  are  quite 
difficult. 

d.  A  slit  is  adjusted  parallel  to    A 
u  plane  mirror,  but  slightly  above 
it.      Any   screen    placed    suitably 
will    receive    now    two    trains    of 

waves ;  one  from  the  slit  directly,  the  other  from  the  vir- 
tual image  of  the  slit  in  the 
mirror. 

—         In  all  these  four  methods 
care   must  be  taken  to   have 
the  two  sources  close  together 
and  accurately  parallel ;  under 
FIG.  273.  these   conditions  a   source  of 


FIG.  272. 


362] 


INTERFERENCE—  DIFFRACTION 


473 


x 


homogeneous  waves  in  front  of  the  first  slit  will  produce 
light  and  dark  interference  bands  on  a  screen  which  is 
parallel  to  the  plane  of  the  two  sources. 

As  stated  in  the  first  chapter,  there  is  a  connection  be- 
tween the  distance  apart  of  the  bands  on  the  screen  and 
the  wave-length  of  the  waves.  This  may  be  best  shown 
by  deducing  the  exact  mathematical  connection. 

Let  A  and  B  be  the  two  identical  sources  of  light,  which 
are  supposed  to  be  very  close  together.  Let  the  screen  be 
parallel  to  the  two  slits  A 
and  B,  and  placed  at  a  con- 
siderable distance  away. 
Draw  a  perpendicular  line 
from  the  opaque  portion  be- 
tween the  slits  to  the  screen, 
and  let  it  meet  it  in  the 
point  C.  A  and  B  are  so 
close  together,  and  C  is  so 
far  away,  that  the  lines  A  0 

and  B  C  may  be  regarded  as  having  the  same  length  as 
the  perpendicular  line,  and  as  being  parallel.  Call  the 
distances  A  B,  c,  and  A  C  (or  B  C),  a.  The  disturbance 
at  any  point,  P,  on  the  screen  is  the  resultant  of  two  dis- 
turbances which  come  from  the  two  slits  A  and  B.  Their 
difference  in  path  may  be  found  by  dropping  a  perpendicu- 
lar, A  D,  from  A  upon  the  line  joining  B  and  P.  Call  the 
distance  from  C  to  P,  C  P,  =  x  ;  then  by  similar  triangles 

AB  =  PC:  AC, 


FIG  2?4 


or  the  difference  in  path, 


ex 
a 


If  this  difference  in  path  is  an  odd  number  of  half 
wave-length,  there  will  be  no  resulting  disturbance  at  P] 
whereas,  if  it  equals  a  whole  number  of  wave-lengths,  the 
effects  will  strengthen  each  other,  and  there  will  be  a 
bright  band. 


474  THEORY  OF  PHYSICS  [CH.  VI 

•1.  Dark  lands. 

The  condition  is  that —  =  (2  n  +  1)  -;  where  n  equals 

a  2i 

any  whole  number  0,  1,  2,  3,  etc.  Corresponding  to  any 
value  of  n  there  will  be  a  definite  value  of  x ;  and  there- 
fore there  will  be  a  succession  of  dark  bands  across  the 
screen,  each  band  being  parallel  to  the  slits. 

The  distance  apart  of  two  bands  is  found  by  giving  n 
two  consecutive  values,  I  and  I  +  1.  Thus  the  distance 

equals  |£  {2  (I  +  1)  +  1  -  (2  I  +  1)}.     Calling  it  D, 

D=a-± w 

2.  Bright  lands. 

c  x 

The  condition  is  that  -  -  =  n  \, 
a 

where  n  equals  in  turn  0,  1,  2,  3,  etc.  Where  n  =  0,  i.  e. 
where  x  =  0,  there  is  a  bright  band  for  all  values  of  X.  All 
waves,  therefore,  no  matter  what  their  wave-numbers,  give 
a  bright  band  at  the  centre  of  the  screen,  directly  opposite 
the  opaque  portion  between  the  two  slits. 

The  distance  apart  of  two  consecutive  bright  bands  is, 
as  before  for  dark  bands, 

2)  =  -\{(b  +  l)-b}  =  a^. 
c  c 

Consequently  the  bright  and  dark  bands  are  equally 
spaced  so  long  as  the  wave-length  is  constant  and  the 
essential  conditions  observed ;  but,  for  waves  of  increasing 
length,  i.e.  of  smaller  wave-number,  the  distances  of  the 
bands  apart  increases. 

It  should  be  noticed  that  the  less  the  distance  between 
the  two  sources,  so  much  the  greater  is  the  distance  apart 
of  the  bands.  Further,  since  D,  a,  and  c  may  all  be  meas- 
ured, this  experiment  gives  a  method  for  the  determination 
of  the  wave-length  of  the  waves.  It  is  not  a  very  accurate 


363] 


INTERFERENCE  —  DIFFRACTION 


475 


Fio.  275. 


method,  however  j  and  so  it  is  not  often  used.  The  method 
now  universally  adopted  for  the  determination  of  wave- 
lengths will  be  explained  later.  (See  Art.  365.) 

363.  Colors  of  Thin  Plates.  Another  phenomenon  due  to 
interference  is  the  so-called  "  colors  of  thin  plates,"  such  as 
are  observed  in  soap-bubbles  or  any  thin  transparent  films. 
It  is  noticed  that  when  "  white 
light"  is  incident  upon  such  a 
transparent  film,  it  appears  col- 
ored both  in  transmitted  and  in 
reflected  light.  The  film  may 
be  a  solid,  a  liquid,  or  a  gas. 
The  explanation  is  not  difficult. 
When  waves  of  any  definite 
wave-number  fall  upon  a  thin 
transparent  film  with  parallel 
faces,  some  of  the  waves  are  re- 
flected at  the  surface ;  others  are  refracted  into  the  film. 
Of  these  waves  refracted  into  the  film,  some  are  reflected 
at  the  lower  surface,  return  to  the  upper  surface  and 
emerge  parallel  to  those  waves  which  were  reflected  on 
incidence;  others,  of  course,  are  refracted  out  at  the  lower 
surface ;  and,  of  those  reflected  back  to  the  upper  surface, 
all  do  not  emerge,  but  some  may  be  reflected  again,  etc. 
It  is  evident  that  the  waves  coming  away  from  the  upper 
surface  consist  of  several  trains  of  waves  all  with  their 
wave-normals  parallel  and  all  of  the  same  wave-number ; 
but  the  separate  trains  have  passed  over  different  paths 
and  have  been  differently  modified.  Consequently,  it  may 
be  that  the  superimposed  waves  are  relatively  retarded ; 
the  crest  of  one  train  may  be  behind  the  crest  of  another. 
This  retardation  depends  upon  three  things :  — 

1.  The  actual  difference  in  the  paths  of  the  different 
waves. 

2.  The  different  velocities  of  the  waves  in  the  film  and 
outside. 


476  THEORY  OF  PHYSICS  [CH.  VI 

3.  The  fact  that  one  train  of  waves  has  been  reflected 
when  passing  from  a  "  faster  "  medium  into  a  "  slower," 
and  the  other  while  passing  just  the  opposite  way.  This- 

introduces  a  difference  which  is  equivalent  to  —  in  the  ap~ 

a 

pearance  of  the  reflected  waves.     (See  Art.  135.) 

If  the  total  retardation  of  one  train  of  waves  behind  the- 
other  is  an  odd  number  of  half  wave-lengths,  the  two- 
waves  will  tend  to  neutralize  each  other.  It  may  be 
shown  that  the  intensities  of  the  two  interfering  trains  of 
waves  are  the  same ;  and,  consequently,  under  these  con- 
ditions there  will  be  absolutely  no  effect  in  the  direction 
of  the  angle  of  reflection. 

For  different  angles  of  incidence  the  retardation  will 
naturally  be  different ;  and  so,  if  the  incidence  of  a  train 
of  waves  of  definite  period  upon  a  thin  film  is  varied, 
there  will  be  angles  for  which  there  will  be  no  reflected 
waves,  and  others  for  which  there  will  be.  The  retarda- 
tion is  different  also  for  plates  of  different  thickness.  The 
transmitted  waves  are  always  such  as  will,  together  with 
the  reflected  waves,  be  equivalent  to  the  incident  waves ;, 
because  there  is  supposed  to  be  no  absorption. 

If,  then,  white  light  is  incident  upon  a  thin  film  at  any 
angle,  there  will  be  some  waves  reflected  (at  the  angle  of 
reflection,  of  course)  ;  but  corresponding  to  this  angle  of 
incidence  and  thickness  of  film,  some  particular  train 
of  waves  will  completely  disappear,  as  just  explained,  from 
the  reflected  waves  ;  and  therefore  the  color-sensation  pro- 
duced by  the  reflected  waves  will  be  complementary  to 
that  which  corresponds  to  the  train  which  has  disappeared. 
(Those  waves  which  do  not  appear  in  the  reflected  waves 
are  refracted  out  at  the  lower  side ;  that  is,  are  trans- 
mitted.) If  the  angle  of  incidence  or  the  thickness  of  the 
film  is  varied,  different  trains  of  waves  will  be  cut  out ; 
and  so  the  color  of  the  reflected  light  will  change. 

If  a  thick  plate  is  used,  instead  of  a  thin  one,  several 


364]  INTERFERENCE  — DIFFRACTION  477 

different  waves  may  be  cut  out  from  the  incident  waves ; 
and  those  which  are  left  may  combine  to  produce  practi- 
€ally  a  white  sensation  ;  so  only  thin  plates  are  colored  by 
the  "  white  light." 

There  is  a  modification  of  this  experiment  which  is 
called  "  Newton's  Kings."  A  film  of  air  is  made  by  plac- 
ing a  convex  lens  upon  a  plane  surface.  There  are  thus 
circular  strips,  as  it  were,  of  varying  thicknesses  ;  and  so,  if 
a  homogeneous  train  of  waves  falls  upon  this  film,  there 
will  be  a  succession  of  thicknesses  of  these  strips,  which 
will  not  allow  the  waves  to  be  reflected.  Consequently 
there  will  be  dark  rings  separated  by  bright  ones.  If 
white  light  is  used,  there  is,  of  course,  a  general  overlap- 
ping of  the  rings  due  to  different  waves,  and  certain  beau- 
tifully colored  rings  appear. 

In  every  case,  of  course,  more  of  the  waves  pass  through 
the  film  than  are  reflected,  because  at  any  transparent 
surface  reflection  is  always  weak.  The  transmitted  effect, 
then,  is  always  complicated  by  the  presence  of  some  waves 
which  have  come  directly  through.  This  is  the  reason 
why  the  transmission  colors  are  always  so  much  feebler 
than  the  reflected  ones. 

364.  Diffraction  by  an  Edge.  In  Chapter  I.,  Article  303, 
it  was  explained  why  the  illumination  near  the  edge  of  a 
shadow  is  complicated  by  the  hiding  of  portions  of  the 
Huyghens'  zones  by  the  obstacle  which  casts  the  shadow. 
This  phenomenon  is  called  "  diffraction." 

As  the  simplest  case,  consider  waves  of  a  definite  wave- 
number  coming  from  a  slit  S ;  and,  after  passing  an  opaque 
obstacle  at  R  with  its  edge  parallel  to  the  slit,  let  them 
fall  upon  a  screen  placed  at  P  T  Q.  The  illumination  at 
different  points  on  the  screen  may  be  studied  by  con- 
structing the  wave-front  at  the  instant  when  it  reaches 
the  obstacle,  i.  e.  when  its  section  is  OR  0',  and  by  draw- 
ing Huyghens'  zones  for  each  of  the  points.  Thus,  the  pole 
of  P  is  0,  where  S  0  Pis  a  radius  of  the  wave-front ;  and 


478 


THEORY  OF  PHYSICS 


[CH.  VI 


Q 


FIG.  276. 

Huyghens'  zones  may  be  constructed  around  0,  as  explained 
in  Chapter  I.  If  the  pole  0  is  near  the  edge  of  the  obstacle, 
that  is,  if  the  point  P  on  the  screen  is  near  the  edge  of  the 
shadow,  it  may  not  be  possible  to  construct  a  large  num- 
ber of  zones  ;  and  so  the  series  m^  —  m»  +  ms  —  m4  +  etc. 
does  not  have  an  infinite  number  of  terms.  If  0  is  so  near 
E  that  there  is  only  a  small  number  of  complete  zones, 
their  effects  at  P  will  almost  completely  neutralize  each 
other,  if  there  is  an  even  number  of  zones ;  while,  if  there 
is  an  odd  number,  the  effect  at  P  will  be  practically  that 
due  to  the  first  zone.  Therefore,  for  small  distances  on 
the  wave-front  away  from  R,  the  edge  of  the  obstacle,  the 
poles  will  be  such  that,  as  they  recede  from  the  edge,  an 
even  number  of  zones  can  be  drawn,  then  an  odd,  then  an 
even,  and  so  on ;  and  there  will,  therefore,  be  on  the  screen 
alternations  in  intensity,  or  bands,  parallel  to  the  edge  of 
the  obstacle  and  the  slit.  These  bands  are,  of  course,  only 
near  the  edge  of  the  shadow,  just  outside  it  in  fact. 

If  Huyghens'  zones  are  constructed  for  any  point,  Q,  on' 
the  screen,  behind  the  obstacle,  the  first  zone*  which  pro- 
duces an  effect  will  not  be  around  its  pole,  Of,  but  at  the 
edge  of  the  obstacle ;  and  it  follows  that  the  entire  effect 
at  Q  is  due  to  one-half  the  first  zone  which  it  has  at  the 
edge  R.  If  Q  is  far  away  from  R,  the  effect  is  less  than  if 


365] 


INTERFERENCE  —  DIFFRACTION 


479 


it  is  near;  and,  consequently,  the  illumination  decreases 
uniformly  for  points  below  the  edge  of  the  obstacle. 

Thus  light  waves  do  actually  bend  in  slightly  behind 
obstacles  ;  but  to  a  minute  degree,  owing  to  the  shortness 
of  the  lengths  of  the  waves. 

These  diffraction  effects  are  observed  near  the  edges  of 
any  shadows  which  are  cast  by  small  sources  of  light ; 
for,  if  the  source  is  large,  the  phenomenon  is  obscured 
by  overlapping. 

365.  Diffraction-Grating.  The  "  diffraction-grating  "  con- 
sists, in  its  simplest  form,  of  a  plate  of  glass  on  which  are 
scratched  by  means  of  a  diamond-point  a  series  of  parallel 
fine  lines  very  close  together  and  at  exactly  regular  inter- 
vals. These  scratches  are,  of  course,  opaque  to  light-waves, 
because  they  scatter  and  diffuse  them ;  but  in  between 
them  there  are  parallel  transparent  [n/rtions.  As  many 
as  15,000  or  20,000  lines  in  a  distance  of  an  inch  (2.54 
cm.)  are  oft  ^  made.  This  ruling  of  a  grating  is  done 
by  means  of  an  automatic  machine,  because  everything 
must  be  }  rfectly  regular. 


FIG.  277. 


If  ordinary  "  white  light "  is  allowed  to  fall  upon  such  a 

ting,  the  emerging  light  appears  colored ;  and  different 

lors  will  be  seen  on  looking  at  the  grating  from  different 

arections.     The  simplest  mode  of  explanation  is  to  show 


480  THEORY  OF  PHYSICS  [CH.  VI 

how,  if  homogeneous  light  is  incident  on  the  grating,  it  is 
only  at  certain  angles  that  any  effect  is  transmitted. 

The  grating  consists  in  principle  of  a  series  of  trans- 
parent slits  placed  parallel  to  each  other  at  regular  inter- 
vals. A  section  of  it  would  then  be  as  shown.  When  a 
homogeneous  train  of  plane  waves  reaches  the  grating,  each 
opening  becomes,  as  it  were,  a  new  source  of  light,  and 
spherical  waves  spread  out  from  each  point.  Consider  the 
•effect  in  any  direction  given  by  a  line  making  an  angle  6 
with  the  normal  to  the  grating.  All  disturbances  in  this 
direction  will  be  brought  to  focus  in  the  focal  plane  of  a 
converging  lens,  if  it  is  placed  so  as  to  intercept  the 
waves ;  and  thus  the  actual  effect  may  be  studied.  The 
distance  from  the  edge  of  one  opening  to  the  correspond- 
ing edge  of  the  other,  a,  is  called  the  "grating-space;" 
and  the  difference  in  path  between  disturbances  going  from 
corresponding  points  in  consecutive  openings  off  in  the 
direction  6  is  evidently  a  sin  6.  Therefore,  if  the  waves 
are  falling  perpendicularly  uppn  the  grating,  so  that  the 
secondary  waves  from  the  grating-openings  leave  at  the 
same  instant,  the  effect  from  any  one  opening  reinforces 
that  from  the  next  one,  if 

a  sin  6  =  n  \, 

where  n  is  any  whole  number  and  X  is  the  wave-length  of 
the  waves ;  and  so,  if  6  satisfies  this  condition,  there  will 
be  a  maximum  effect  produced  at  the  corresponding  point 
in  the  focal  plane  of  the  interposed  lens.  There  will  thus 
he  points  of  maximum  effect  in  the  focal  plane,  correspond- 
ing to  successive  values  of  n ;  and,  of  course,  the  effect  will 
be  symmetrical  on  the  two  sides  of  the  axis,  for  6  can  be 
drawn  below  the  axis  as  well  as  above  it. 

The  general  appearance  on  the  focal  plane  may  be  rep- 
resented as  in  the  figure,  where  the  elevations  mean 
maximum  effects  or  illumination.  If  there  is  a  great 
number  of  lines  in  the  grating,  say  50,000  or  100,000, 


365] 


INTERFERENCE  —  DIFFRACTION 


481 


FIG.  278. 

these  curves  for  maximum  illumination  contract  until,  in- 
stead of  there  being  a  wide  illumination  around  each 
position  given  by  0 ,  there  is  only  practically  a  straight 
line  (if  the  waves  come  from  a  slit  originally).  Under 
these  conditions  it  is  possible  to  measure  6  accurately. 
Thus,  in  the  formula 


a  sin  9  =  n  \ 


(2) 


a,  the  grating-space,  may  be  measured ;  so  can  6 ;  and  n, 
the  "  order  "  of  the  spectrum,  is  some  known  whole  num- 
ber, 1,  2,  3,  etc.  Consequently  X,  the  wave-length  of  the 
waves,  may  be  measured  most  accurately. 

In  the  accompanying  table  are  given  some  values  of  the 
wave-lengths  of  certain  absorption  lines  in  the  solar  spec- 
trum, which  have  received  definite  names. 

TABLE   XVII 
WAVE-LENGTHS  IN  CENTIMETRES 


K 

.  .  .  0.00003933825 

D2 

.  .  .  0.00005890186 

H 

.  .  .  0.00003968625 

Di 

.  .  .  0.00005896357 

g 

.  .  .  0.00004226904 

C 

.  .  .  0.00006563054 

G 

.  .  .  0.00004340634 

B 

.  .  .  0.00006870186 

F 

.  .  .  0.00004861527 

hi 

.  .  .  0.00005183791 

E2 

.  .  .  0.00005269723 

Bi 

.  .  .  0.00005270495 

The  unit  commonly  adopted  for  expression  of  wave-lengths 
is  not  the  centimetre;  but  0.00000001   of  a  centimetre. 

16 


482  THEOKY   OF  PHYSICS  [CH.  VI 

Such  a  unit  is  called  an  Angstrom  unit,  from  the  name  of 
the  great  Swedish  spectroscopist  who  first  suggested  its  use. 
Thus,  the  wave-length  of  D\  is  5896.357  Angstrom  units. 

Since,  then,  for  any  value  of  X  there  is  a  succession  of 
definite  values  of  6  for  which  there  will  be  maximum 
effects ;  if  white  light  is  used,  the  different  trains  of  waves 
will  be  spread  out  separately,  and  colored  effects  will  be 
produced.  It  is  also  evident  that  a  grating  may  take  the 
place  of  a  prism  for  the  examination  of  the  spectra  of 
different  sources ;  and  so  grating-spectroscopes  are  often 
used. 

It  should  further  be  noted  that  the  deviation  and  dis- 
persion produced  by  a  grating  has  not  the  faintest  connec- 
tion with  its  material ;  they  are  the  same  for  all  gratings 
of  the  same  number  of  lines  per  inch. 

Such  a  grating  as  has  been  just  described  is  called  a 
"  transmission "  one  ;  because  the  waves  are  transmitted 
through  its  openings.  Keflecting  gratings  may  also  be 
used,  where  an  opaque  mirror,  e.  g.  polished  metal,  has 
a  series  of  parallel  scratches  upon  it  at  regular  intervals. 
Again,  instead  of  having  the  rulings  on  a  plane  surface, 
they  may  be  on  a  curved  surface ;  and  concave  reflecting 
gratings  have  certain  most  interesting  properties,  which 
cannot,  however,  be  discussed  here. 


CHAPTEE  VII 


DOUBLE   REFRACTION 

366.  As  explained  in  Chapter  III.,  when  a  train  of 
waves  in  any  medium  falls  upon  a  surface  which  separates 
that  medium  from  another,  some  waves  enter  the  second 
medium,  and  have  a  definite  wave-front.  This  fact  is  true 
of  all  isotropic  media  ;  but  it  is  observed  that,  when  waves 
pass  into  certain  crystals  in  certain  directions,  there  are 
two  refracted  wave-fronts.  Such  bodies  as  this  are  called 
"  doubly  refracting." 

If  plane  waves  fall  upon  a  plate  of  a  doubly  refracting 
substance,  which  has  parallel  sides,  there  will  be,  in  gen- 
eral, two  trains  of  plane  waves 
produced  in  the  plate,  in 
slightly  different  directions  ; 
and  these  two  waves  will 
emerge  with  their  normals  in 
the  same  direction  as  the  in- 
cident waves.  Following  the 
path  of  any  one  disturbance, 
it  is  seen  to  produce  two  "rays." 
If  a  black  point  is  placed  on 
one  side  of  the  plate,  there  will 
be  two  virtual  images  of  it  at 
different  points  in  the  plate  produced  by  the  two  sets  of 
waves ;  and  so,  if  the  plate  is  viewed  from  the  opposite 
side,  two  points  will  be  apparently  seen.  (Compare  the 
similar  problem  in  a  singly  refracting  plate,  Art.  324.) 


FIG.  279. 


484  THEORY  OF  PHYSICS  [CH.  VII 

It  is  found  by  experiment  that  all  crystals,  with  the 
exception  of  those  which  belong  to  the  cubical  system,  are 
doubly  refracting ;  and  that  isotropic  substances  become  so 
if  they  are  subjected  to  unequal  strains,  e.  g.  glass  un- 
evenly annealed.  In  certain  crystals  there  is  one  direc- 
tion in  which  waves  may  pass  through  without  suffering 
double  refraction  ;  such  are  called  "  uniaxial."  In  other 
crystals  there  are  two  such  directions ;  and  such  are  called 
"  biaxial." 

367.  Uniaxial  Crystals.      The  best   known  examples  of 
these  are  quartz  and  Iceland  spar;  and  both  are  easily 
obtained  and  studied.     It  is  found  by  experiment  that  of 
the  two  waves  produced  in  general  in  a  uniaxial  crystal 
by  a  single  incident  train  of  waves,  one  obeys  the  laws  of 
ordinary  refraction,  the  other  does  not.     The  first  train 
of  waves  is  called  "  the  ordinary  wave  ;  "  the  second,  "  the 
extraordinary  ; "  and  the  latter,  as  a  rule,  obeys  neither  of 
the  two  laws  of  refraction.     If  a  plate  of  Iceland  spar  is 
placed  over  a  black  point,  two  points  are  seen,  as  explained 
above ;  but,  if  now  the  plate  is  turned  around  in  its  own 
plane,  the  image  due  to  the  extraordinary  waves  will  re- 
volve around  that  due  to  the  ordinary  waves,  which,  of 
course,  remains  at  rest. 

368.  Biaxial  Crystals.     In  biaxial  crystals  neither  train 
of  waves  obeys  the  ordinary  laws  of  refraction  in  general, 
although  for  particular  sections  of  the  crystal  one  or  both 
may.     So  in  these  crystals  both  trains  of  waves  are  extra- 
ordinary. 

369.  Since  in  a  doubly  refracting  crystal  the  two  trains 
of  waves  have  different  wave-points,  it  is  evident  that  they 
travel  with  different  velocities,  and  so  have  different  in- 
dices of  refraction.    (Read  Art.  319.)    Therefore,  if  a  prism 
is  made  of  a  doubly  refracting  substance,  and  if  homogene- 
ous plane  waves  fall  upon  its  face,  two  plane  waves  will 
emerge,  in  different  directions.      If  a  converging  lens  is 
interposed,  there  will  be  two  images  in  the  focal  plane 


371] 


DOUBLE  REFRACTION 


485 


corresponding  to  the  two  directions  of  the  waves.     Thus 
the  two  trains  of  waves  may  be  separated,  and  studied. 

370.  Nicol's  Prism.  In  Iceland  spar  it  may  be  proved 
that  the  index  of  refraction  of  the  ordinary  waves  is  greater 
than  that  of  the  extraordinary.  A  most  interesting  appli- 
cation of  this  fact  has  been  made  so  as  to  completely  sepa- 
rate the  extraordinary  and  ordinary  waves.  In  speaking  of 
refraction  (Art.  320),  it  was  shown  that  if  waves  were 
passing  from  a  medium  of  greater  refractive  index  to  one 
of  less  (i.  e.  of  less  velocity  to  greater),  total  reflection  re- 
sulted if  the  angle  of  incidence  exceeded  a  certain  critical 
angle.  It  has  been  discovered  that  the  index  of  refraction 
of  Canada  balsam  is  intermediate  between 
those  of  the  extraordinary  and  ordinary  waves. 
A  piece  of  Iceland  spar  may  then  be  cut  in 
two  and  gummed  together  again  with  Canada 
balsam ;  so  that  when,  as  a  result  of  a  single 
train  of  incident  waves,  the  extraordinary  and 
the  ordinary  waves  fall  upon  the  surface  of 
the  Canada  balsam,  the  angle  of  incidence  is 
such  that  it  exceeds  the  critical  angle  for  the 
ordinary  waves,  and  so  they  will  be  totally 
reflected.  The  extraordinary  waves,  on  the 
contrary,  pass  directly  through  the  trans- 
parent balsam,  losing  very  little  light  by  re- 
flection, and  emerge  on  the  farther  side  of  the 
piece  of  Iceland  spar.  Thus  the  two  waves 
are  completely  separated.  Ordinarily,  the 
sides  of  the  Iceland  spar  are  blackened  so  as  to  absorb 
the  ordinary  waves  when  they  strike  them ;  and  thus 
only  extraordinary  waves  can  pass  through.  Such  an 
arrangement  as  described  is  called  a  "  Nicol's  prism  ; "  or 
more  often,  a  "Nicol." 

371.  Property  of  Tourmaline.  The  uniaxial  crystal  tour- 
maline is  doubly  refracting ;  but,  if  its  thickness  exceeds 
more  than  1  or  2  mm.,  only  the  extraordinary  waves 


FIG.  280. 


486  THEORY  OF  PHYSICS  [CH.  VII 

emerge,  as  the  ordinary  ones  are  absorbed.  Consequently, 
if  ordinary  light  'falls  upon  tourmaline,  only  one  kind  of 
waves  emerge.  Tourmaline  is,  however,  a  rather  opaque 
crystal ;  and  not  much  light  is  transmitted  in  any  case. 
There  are,  however,  other  substances,  more  transparent, 
which  have  this  same  selective  absorption. 


CHAPTEK    VIII 

POLARIZATION 

So  far,  in  the  discussion  of  the  phenomena  associated 
with  Light,  explanations  have  been  given  in  terms  of  the 
wave-theory ;  but  nothing  has  been  said  as  to  the  nature 
of  the  waves  themselves.  No  phenomenon  so  far  de- 
scribed requires  the  waves  to  be  transverse,  or  longitu- 
dinal ;  but  now  certain  phenomena  will  be  discussed  which 
do  definitely  require  for  their  explanation  that  the  ether- 
waves  are  transverse.  These  phenomena  are  generally 
classed  under  the  name  of  "  polarization,"  for  reasons  that 
will  soon  appear» 

372.  Polarization  by  Reflection.  If  a  train  of  plane  waves 
fall  upon  a  plane  piece  of  glass,  or  other  transparent  ma- 
terial, there  will  be  both  a  reflected  and  a  refracted  train 
of  waves  ;  and,  if  this  reflected  train  is  made  to  fall  upon 
a  second  plane  piece  of  glass  parallel  to  the  first,  there  will 
also  be  a  reflected  and  a  refracted  train  in  general.  It  is 
noticeable,  though,  that  there  are  marked  differences  in 
the  intensities  of  the  reflected  and  the  refracted  trains  at 
this  second  mirror  ;•  the  latter  being  much  the  weaker.  If 
the  angle  of  incidence  of  the  first  train  of  waves  on  the 
first  mirror  is  varied,  an  angle  may  be  found  for  which 
there  is  absolutely  no  refracted  train  at  all  at  the  second 
glass  mirror.  If  now  the  second  mirror  is  revolved  about 
an  axis  parallel  to  the  wave-normal  of  the  waves  incident 
upon  it,  the  angle  of  incidence  will  not  change ;  but  the 
plane  of  incidence,  i.  e.  the  plane  including  the  normal  to 


488  THEORY  OF  PHYSICS  [CH.  VIII 

the  surface  and  the  normal  to  the  wave-front,  will.  When 
the  plane  of  incidence  has  been  thus  revolved  through  90°, 
absolutely  no  waves  at  all  are  reflected  from  the  second 
mirror,  but  all  the  incident  waves  are  refracted. 

How  is  it  possible  to  account  for  this  fact,  that  waves 
reflected  from  one  surface  and  falling  upon  another  cannot 
be  reflected  ?  No  explanation  can  be  given  if  the  ether- 
waves  are  longitudinal ;  for  it  is  impossible  to  think  of  any 
modification  in  them  which  could  prevent  their  reflection 
at  the  second  surface.  If  the  ether-waves  are  transverse, 
the  explanation  is  not  difficult.  On  this  theory  a  train  of 
ordinary  ether-waves  consists  of  the  advance  of  some  vibra- 
tion which  is  perpendicular  to  the  path  of  the  waves.  The 
vibration  of  any  individual  particle  must  be  in  the  wave- 
front,  but  may  be  of  any  periodic  nature ;  it  may  be  in  a 
straight  line  in  any  direction,  it  may  be  in  a  circle,  it  may 
be  in  an  ellipse  ;  and  ordinary  ether-waves  may  be  re- 
garded as  a  varying  mixture  of  all  these  vibrations.  But 
when  such  a  train  of  waves  reaches  the  surface  of  the 
glass  at  the  definite  angle  of  incidence,  it  may  be  supposed 
that  only  particular  vibrations  are  reflected.  There  is 
almost  conclusive  evidence  that  with  ordinary  ether-waves 
only  those  vibrations,  or  components  of  vibrations,  are 
reflected  which  are  perpendicular  to  the  plane  of  incidence, 
i.  e.  along  the  surface.  (The  other  vibrations,  i.  e.  those 
which  are  in  the  plane  of  incidence,  and  so  strike  down,  as 
it  were,  into  the  surface,  are  refracted.)  The  plane  surface 
of  glass,  then,  for  this  angle  of  incidence  allows  only  those 
vibrations  to  be  reflected  which  are  parallel  to  the  surface. 
All  the  vibrations  in  the  reflected  wave  are  in  the  same 
direction ;  viz.,  parallel  to  a  line  perpendicular  to  the  plane 
of  incidence.  Such  a  train  of  waves  is  said  to  be  "  plane 
polarized  in  the  plane  of  incidence."  When  this  train  of 
waves  falls  upon  the  second  glass  mirror  placed  parallel  to 
the  first,  its  angle  of  incidence  and  plane  of  incidence  are 
the  same  as  they  were  for  the  first  mirror ;  and  therefore 


373] 


POLARIZATION 


489 


all  the  waves  will  be  reflected,  the  vibrations  being  parallel 
to  the  surface. 

But,  if  the  second  mirror  is  so  turned  that  the  plane  of 
incidence  is  at  right  angles  to  that  for  the  first  mirror,  the 
vibrations  in  the  incident  waves  are  in  the  plane  of  inci- 
dence, and  so  enter  the  surface ;  consequently,  since  for 
this  angle  of  incidence  the  surface  reflects  only  those  vi- 
brations which  are  parallel  to  the  surface,  there  is  no 
reflected  train  of  waves,  all  the  incident  waves  are  refracted. 

This  particular  angle  of  incidence  for  which  a  plane 
transparent  substance  will  reflect  only  vibrations  parallel 
to  its  surface  is  called  the  "angle  of  polarization."  In 
most  cases,  at  the  polarizing 
angle  only  a  maximum  amount 
of  the  waves  whose  vibrations 
are  along  the  surface  is  reflected; 
that  is,  the  polarization  is  not 
complete.  But  for  those  sub- 
stances whose  indices  of  refrac- 
tion are  in  the  neighborhood  of 
1.46,  the  polarization  is  complete 
at  the  polarizing  angle.  It  is 
found  by  experiment  that  this 
angle  of  incidence  is  such  that  the  reflected  and  refracted 
wave-normals  are  at  right  angles  to  each  other.  If  a  is  the 
angle  of  polarization,  it  follows  that  the  angle  of  refraction 
corresponding  to  it  is  90°  —  a.  Hence  if  ^  is  the  index  of 
refraction, 

sin  a 


FIG.  281. 


or 


sin  (90°  -  a) 
tan  a  =  u 


Even  if  the  angle  of  incidence  is  not  the  polarizing 
angle,  there  is  always  a  proportion  of  the  incident  waves 
plane  polarized  by  the  reflection. 

373.   Polarization  by  Doubly  Refraction.     If  the  two  trains 


490  THEORY  OF  PHYSICS  [CH.  VIII 

of  plane  waves  produced  in  a  doubly  refracting  sub- 
stance are  allowed  in  turn  to  fall  upon  a  transparent 
mirror  at  its  polarizing  angle,  it  may  be  proved  that 
they  are  both  plane  polarized,  but  that  they  are  polar- 
ized in  planes  at  right  angles  to  each  other.  That  is, 
if  the  mirror  is  so  placed  as  to  completely  reflect  one 
train  of  waves,  the  other  will  be  completely  refracted ; 
and,  if  the  mirror  is  turned  around  an  axis  parallel  to 
the  wave-normal  of  the  incident  waves  until  its  plane 
of  incidence  has  revolved  90°,  the  waves  which  were 
refracted  before  are  now  reflected  and  vice  versa.  This 
proves,  then,  that  the  vibrations  in  the  two  trains  of  waves 
are  in  straight  lines  at  right  angles  to  each  other. 

This  fact  furnishes  at  once  a  most  convenient  method  for 
producing  plane  polarized  waves.  For  a  Nicol's  prism  or  a 
piece  of  tourmaline  allows  only  one  train  of  waves  to  pass 
through ;  and  consequently  in  the  emerging  waves  the  vi- 
brations are  all  in  one  direction.  There  is,  then,  in  a  Nicol 
a  certain  fixed  direction  along  which  vibrations  must  be  in 
order  to  be  transmitted. 

If  waves  emerging  from  one  Nicol  enter  a  second,  they 
can  emerge  if  the  direction  of  the  vibrations  in  the  in- 
cident waves  is  that  direction  which  the  second  Nicol  can 
transmit,  or  if  the  vibration  can  have  a  component  in 
that  direction.  If  the  Nicols  are  so  placed  that  the  two 
fixed  directions  are  parallel,  all  the  incident  waves  will  be 
transmitted,  and  the  Nicols  are  said  to  be  "  parallel."  If, 
on  the  other  hand,  the  two  fixed  directions  are  at  right 
angles  to  each  other,  no  waves  will  be  transmitted,  and 
the  Nicols  are  said  to  be  "  crossed." 

It  should  be  observed  that,  since  it  is  the  extraordinary 
wave  which  is  transmitted  by  a  Nicol,  the  vibrations  in 
this  wave  must  be  parallel  to  the  fixed  direction  in  the 
Nifiol;  the  vibrations  in  the  ordinary  wave  are  at  right 
angles  to  this  direction.  Consequently,  when  two  Nicols 
are  crossed,  the  vibrations  which  emerge  from  the  first  as 


X 

x 

/ 


\ 
\ 
\ 

\ 
\ 


V 


373]  POLARIZATION  491 

an  extraordinary  wave  become  the  ordinary  wave  of  the 
second,  and  are  totally  reflected  at  the  layer  of  Canada 
balsam. 

This  fact  also  explains  immediately  an  experiment  of 
Huyghens',  which  consisted  in  looking  at  any  small  source 
of  light  through  two  plates  of  Iceland  spar  cut  exactly  alike 
and  placed  over  each  other.  In  general  four  images  are 
seen ;  but,  as  one  plate  is  turned  around  an  axis  perpen- 
dicular to  itself,  there  are  four  positions  90°  apart  for 
which  only  two  images  are  seen. 
The  vibrations  coining  through  the  \ 

first  plate  are  at  right  angles  to 
each  other,  e.  g.  along  a  and  b ;  but 
now,  falling  upon  the  second  plate, 
in  which  there  are  only  two  direc-  / 

tions  which  vibrations  can  have, 
e.  g.  a'  and  I',  the  vibration  along 
a  will  have  components  along  a1  Fl  2«2 

and  b',  and  so  will  the  vibration 

along  b.  Consequently  there  will  be  four  emerging  waves 
in  general.  (The  vibrations  emerge  in  two  directions  only  ; 
but,  as  the  velocities  of  the  two  kinds  of  vibrations  in  the 
two  plates  are  different,  there  will  in  general  be  four  wave- 
fronts.)  But  if  the  two  sets  of  perpendicular  lines  are 
parallel,  there  will  obviously  be  only  two  emerging  waves  ; 
and,  as  one  plate  is  turned  over  the  other,  it  will  happen 
four  times  in  one  complete  revolution  that  these  lines  are 
parallel. 

If  a  plane  polarized  wave  falls  upon  a  thin  plate  of  a 
doubly  refracting  substance,  its  vibrations  will  be  resolved 
into  two  at  right  angles  to  each  other,  because  a  doubly 
refracting  substance  allows  in  its  transmitted  waves  only 
two  directions  of  vibrations,  which  are  at  right  angles,  and 
which  correspond  to  the  two  waves  which  characterize 
double  refraction.  It  has  been  explained  in  the  last  chap- 
ter how  double  refraction  is  due  to  the  fact  that  the  two 


492  THEORY  OF  PHYSICS  [CH.  VIII 

sets  of  waves  produced  by  it  travel  with  different  veloci- 
ties. Consequently,  when  the  plane  polarized  wave  is 
resolved  by  the  doubly  refracting  plate  into  two  waves 
whose  vibrations  are  at  right  angles  to  each  other,  these 
two  waves  will  travel  through  the  plate  with  different 
velocities ;  and  one  will  emerge  in  advance  of  the  other. 
The  actual  vibration  in  the  emerging  wave  will  be  then  a 
combination  of  two  vibrations  in  straight  lines  at  right 
angles  to  each  other,  one  vibration  being  in  general  a  trifle 
farther  advanced  than  the  other,  i.  e.  one  vibration  may  be 
at  the  end  of  its  path  when  the  other  is  just  passing 
through  its  central  position,  etc. 

374.  Circularly  Polarized  Waves.  If  the  thickness  of  the 
doubly  refracting  plate  is  such  as  to  make  one  vibration 
come  through  in  a  time  faster  than  the 
other  by  one  quarter  of  a  period,  the 
vibration  in  the  emerging  wave  is  clue 
to  a  combination  of  one  vibration  which 
is  at  the  end  of  its  path,  x,  when  the 
other  is  just  passing  through  the  middle 
of  its  path,  6>,  towards  y.  The  actual 
path,  or  vibration,  will,  then,  be  an  el- 
FIG.  283.  lipse  whose  axes  are  the  lines  which 

mark  the  direction  and  extent  of  the 
component  vibrations.  The  emerging  waves  are  said  to 
be  "  elliptically  "  polarized. 

If,  however,  these  two  axes  are  of  equal  length,  the 
vibration  will  be  a  circle ;  and  the  waves  are  said  to  be 
"  circularly  "  polarized.  If  the  two  axes  are  of  the  same 
length,  this  means  that  the  two  component  waves  are  of 
equal  intensities  ;  and  this  condition  may  be  easily  se- 
cured. For,  if  the  direction  of  vibration  of  the  incident 
plane  polarized  waves  makes  an  angle  of  45°  with  either 
one  of  the  fixed  directions  of  vibration  in  the  doubly  re- 
fracting plate,  the  incident  vibrations  will  be  resolved  into 
two  equal  components  along  those  two  directions. 


376]  POLARIZATION  493 

A  plate  of  such  a  thickness  as  to  produce  a  difference  of 
time  of  a  quarter  of  a  period  is  called  a  "  quarter-wave 
plate."  So,  in  order  to  produce  circularly  polarized  waves, 
it  is  simply  necessary  to  allow  ordinary  plane  waves  to 
pass  through  a  Nicol  and  then  through  a  quarter-wave 
plate,  the  direction  of  vibration  in  the  Nicol  making  an 
angle  of  45°  with  the  directions  of  possible  vibration  in 
the  plate. 

Conversely,  if  circularly  polarized  waves  are  passed 
through  a  quarter-wave  plate,  they  become  plane  polar- 
ized, because  the  plate  makes  one  of  the  component  vibra- 
tions into  which  the  circular  vibration  may  be  resolved  lag 
behind  the  other  quarter  of  a  period ;  and  two  vibrations  at 
right  angles  to  each  other,  which  pass  through  o  together,  or 
which  reach  x  and  y  at  the  same  instant,  will  combine  to 
produce  a  vibration  in  a  straight  line.  If  a  train  of  waves 
is,  then,  circularly  polarized,  the  fact  may  be  proved  by 
seeing  whether  it  is  possible  so  to  turn  a  Nicol  that  it 
prevents  any  waves  passing  after  the  train  of  waves  has 
traversed  a  quarter-wave  plate. 

375.  Interference  of  Polarized  Waves.     Two  waves  cannot 
completely  interfere  unless  their  vibrations  are  in  the  same 
direction ;  and,  if  any  one  of  the  interference  experiments 
in  Article  362  is  modified  by  placing  a  Nicol's  prism  in 
front  of  each  of  the  two  sources  of  light  so  that  the  two 
interfering  waves  are  plane  polarized,  it  may  be  proved 
that  interference-bands  do  not  occur  when  the  two  Nicols 
are  crossed.     If  they  are  parallel,  the  bands  have  a  maxi- 
mum intensity ;  while  in  intermediate  positions  the  bands 
are  feebler. 

This  experiment  proves  conclusively  that  ether-waves 
are  transverse. 

376.  Colors  Due  to   Polarization.      When   homogeneous 
waves  are  passed  through  a  thin  plate  of  a  doubly  refract- 
ing substance,  there  are  two  emerging  sets  of  vibrations, 
at  right  angles  to  each  other ;  and  one  set  is  retarded  be- 


494  THEORY  OF   PHYSICS  [CH.  VIII 

hind  the  other,  as  explained  in  Article  374.  These  two 
waves  cannot  interfere,  because  the  vibrations  are  at  right 
angles  to  each  other ;  but,  if  these  waves  now  pass  through 
a  Nicol's  prism,  they  will  each  produce  a  train  of  waves 
whose  vibrations  are  in  the  same  direction,  viz.,  the  fixed 
direction  of  the  Nicol,  the  initial  vibrations  being  resolved 
into  components  in  this  direction  and  at  right  angles  to  it, 
the  latter  not  emerging.  Consequently,  the  two  emerging 
waves  are  now  in  a  condition  to  interfere.  If  the  waves 
which  are  incident  on  the  doubly  refracting  plate  are  ordi- 
nary waves,  the  mode  of  vibration  in  them  will  be  con- 
stantly changing ;  and  so  these  two  waves  which  emerge 
from  the  Nicol  will  not  have  any  constant  character.  If, 
however,  the  incident  waves  are  plane  polarized  by  being 
passed  through  a  Nicol,  the  two  emerging  waves  will  have 
definite  intensities.  One  of  the  waves  will  be  retarded  in 
the  doubly  refracting  plate  behind  the  other ;  and  it  may 
be  easily  proved  that,  for  definite  thicknesses  of  the  plate, 
the  two  sections  of  the  homogeneous  waves  will  com- 
pletely interfere.  For  waves  of  different  wave-numbers 
different  thicknesses  of  the  plate  will  serve  to  absorb  them. 
Consequently,  if  "white  light"  passes  through  a  Nicol, 
then  through  a  thin  doubly  refracting  plate,  and  finally 
enters  a  second  Nicol,  a  certain  definite  color  will  be  ab- 
sent; and  so  its  complementary  color  will  be  seen.  (This 
is  true  except  for  four  positions  of  the  two  Nicols.)  If  the 
doubly  refracting  plate  is  of  varying  thickness,  there  will 
be  a  definite  color  corresponding  to  each  thickness.  The 
colors  observed  when  the  Nicols  are  in  any  one  position  are 
complementary  to  those  observed  when  the  position  of  one 
Nicol  is  turned  through  90°. 

If  the  Nicols  are  crossed,  no  light  at  all  would  come 
through  them  if  the  doubly  refracting  plate  were  not  in- 
terposed between  them.  This  fact  of  the  appearance  of 
color-phenomena  when  a  thin  plate  of  doubly  refracting 
substance  is  placed  between  two  crossed  Nicols  is  the  best 


377]  POLARIZATION  495 

test  there  is  for  the  detection  of  double  refraction  in  any 
substance. 

There  are  many  other  ways  of  producing  colored  phe- 
nomena by  means  of  polarized  light-waves,  but  they  are 
too  complicated  for  discussion  here. 

377.  Rotation  of  the  Plane  of  Polarization.  As  already 
noted  in  ELECTRICITY  (Art.  295),  when  a  plane  polarized 
wave  is  passed  along  the  lines  of  force  in  a  magnetic  field, 
the  plane  of  polarization  is  rotated ;  that  is,  the  direction 
of  vibration  changes. 

Many  natural  substances  produce  rotation  of  the  plane 
of  polarization  also,  —  quartz,  if  cut  with  its  faces  perpen- 
dicular to  its  axis  ;  and  many  "  active  "  organic  chemicals, 
such  as  some  tartaric  acids  and  some  sugars  in  aqueous  so- 
lution. In  certain  cases  the  plane  is  rotated  towards  the 
right,  in  others  towards  the  left;  but  in  all  these  sub- 
stances, if  the  plane  polarized  waves  are  made  to  retrace 
their  paths  by  being  sent  back  in  the  opposite  direction, 
exactly  the  opposite  change  takes  place  ;  the  rotation,  as 
it  were,  untwists. 

This  last  fact  is  not  true  of  the  rotation  produced  in  a 
magnetic  field ;  because  there,  entirely  independently  of 
whether  the  waves  are  in  the  same  or  the  opposite  direc- 
tion along  the  lines  of  force,  the  rotation  is  always  con- 
nected with  the  direction  of  the  lines  of  force  by  the 
"right-handed  screw  law." 

It  should  be  noted  that  in  both  cases  the  amount 
of  the  rotation  is  different  for  waves  of  different  wave- 
numbers,  being  least  for  waves  of  the  least  wave-numbers. 


INDEX 


NUMBERS  REFER  TO  PAGES 


Aberration,  Chromatic 433, 442 

,  Spherical 420 

Absolute  Temperature 214,  263 

Zero 264 

Absorption  of  Energy 93,  249,  250,  465 

,  Surface 465 

Acceleration,  Angular 22 

,  Linear 18 

,  Composition  and  Resolution 19,  20,  22,  27,  29 

Due  to  the  Earth,  "  g  " 37,  38,  96,  97,  101 

Achromatism 458 

Activity 82 

Air-Pumps 137-140 

Alternating  Dynamo 379 

Ampere  (Unit  of  Electric  Current) 368 

Ampere's  Theory  of  Magnetism 362 

Amplitude 28,  30,  147 

Angle,  Numerical  Measure  of 22 

,  Critical 427 

,  Polarizing 489 

of  Incidence 408 

of  Minimum  Deviation 430 

of  Reflection • 409 

of  Refraction 424 

Angstrom  Unit  for  Wave-Leugths 482 

Anode 316 

Arago's  Disc 374 

Archimedes' Principle 117-119 

Arc-Lamp 330 

Armature  of  Dynamo 381 

Astatic  Magnetic  Needle •  .     .     .     332 

Atmospheric  Moisture     . 238 

Pressure  129 


498  INDEX 

Atomic  Weight , 316 

Atomizer 134 

Atwood's  Machine 39 

Avogadro's  Law 258  et  seq. 

Axis  of  Lens 441 

of  Rotation 14,  21 

Balance,  Chemical .     .       84 

Ball-Nozzle 122 

Barometer 130 

,  Correction  of 130,  208 

Beats 163,  189 

Bells,  Vibrations  of 181 

Biprism 472 

Boiling-Point 234 

,  Effect  of  Dissolved  Substances  on 238 

, of  Pressure  on 235 

Boyle's  Law 135,  212,  257 

Bulk  Modulus 105,  153,  183 

Calorie ' 216 

Calorimetry 220 

Capacity,  Electric 299 

Capillarity 123-128 

Capillary  Surfaces 126,  128 

Tubes 126 

Cathode.     See  Kathode 316 

Caustic  Surface 421 

Centigrade  Scale  of  Temperature 202 

Centimetre 7 

Centrifugal  Eorce 40 

Charles'  Law 212,  258 

Chemical  Equivalent 316 

Formulae 260 

Chromatic  Scale 191 

Circle,  Uniform  Motion  in  a 25 

Clark  Cell,  Standard  of  E.  M.  F 315,368 

Collimator 461 

Color 469 

,  Absorption 4G6,  469 

,  Complementary 468 

,  Connection  between  Wave-Number  and 397 

,  Polarization 493 

,  Surface 469 

Sensation 468,  470 

Colors  of  Thin  Plates  .     475 


INDEX  499 

Combination,  Chemical 196,  243 

,  Heat  of 243 

Combination- Notes 165 

Combustion 196 

Commutator  of  Dynamo 382 

Compressibility  of  a  Gas 135 

of  a  Liquid 109 

of  a  Solid.     .     . 104 

Compression-Pump 140 

Condensation  of  Vapors  .     .     .     .• 233,  239 

Cone  of  Waves v 412 

Condensers,  Electric  .     .     .     .     .     . 294-297 

— , ,  Capacity  of   . 299 

, ,  Discharge  of 297,  304 

, , ,  Secondary 298 

, ,  Energy  of 302 

Conductance,  Electric 334 

Conduction,  Electric 304  et  seq. 

,  Heat 246 

Conductors,  Electric 274,275 

, ,  Nature  of ; 315,326 

Convection  of  Heat 245 

Co-planar  Forces 61 

Cords,  Vibrations  of 173-175 

Coulomb,  Unit  of  Electricity 368 

Coulomb's  Law  of  Electrostatic  Force 273 

Law  of  Magnetic  Force 346 

Couple .-.-.-.- 67 

Critical  Angle 427 

Temperature,  State,  etc •  ..    , 242 

Crookes'  Radiometer  . 254 

Cryohydrates 230 

Crystals,  Expansion  of 205 

,  Biaxal 484 

,  Uniaxal 484 

Curvature,  Measure  of 413 

Dalton,  Law  of  Mixtures 135,  238,  258 

Daniels'  Cell 315 

Declination,  Magnetic 356 

Density » 8 

of  Water 208 

,  Measurement  of,  of  a  Gas 134 

, ,  of  a  Liquid   . 115,116,118 

, ,  of  a  Solid 118 

Depolarization 494 


500  INDEX 

Deviation 429,  455 

,  Minimum 430 

Dew  —  Dew-Point 238 

Diamagnetic  Bodies 341 

Diamagnetism,  Weber's  Theory  of 385 

Diatonic  Scale 190 

Dielectric  Constant 276,  300 

Dielectrics ...    ........     .....    .     274 

Differential  Notes  .     . ,    .     .     .     165 

Diffraction  around  an  Edge 404,  477 

Diffraction-Grating    .     .     .     . 479 

Diffusion  of  Gases 134 

of  Light •'•;.-.  .-.'•..    . ..-..    ...     391 

of  Liquids 122 

Dip,  Magnetic 357,  376 

Discord      •    y    • 188 

Dispersion 455 

,  Anomalous 457,  465 

Displacement  (Mechanical) 14 

Dissociation .,    .     ....     .     242,244,318,322 

,  Heat  of " 243 

Divided  Circuits,  Laws  of 336 

Doppler's  Principle 160 

Dulong  and  Petit's  Law 220 

Dynamics 31  et  seq. 

Dynamo,  Alternating 379 

,  Gramme-Ring 381 

Dyne 36 

Ear « 158 

Earth-Inductor :...-••• 375 

Echo     .     .     .•.-.-.     .     .    .     .    .    .  ~.    . 166 

Eclipses 405 

Efficiency 263 

Efflux  of  Gases .'....  134 

of  Liquids 121 

Effusion ... 134 

Elasticity.     .     .     .    v-  '"._....;     .    .     . 3,  102  et  seq. 

• ,  Coefficient  of ._ 104 

,  Limit  of 103 

of  a  Gas 136 

Electricity,  Distribution  of      .     .     .     .     ...........     ....       286,  300 

,  Positive  and  Negative « 270 

,  Quantity  of 272 

,  Specific  Attraction  of  Matter  for      .     .     ..  ;. 271 

,  Surface-Density  of 286 


INDEX  501 

Electricity,  Units  of,  Electro-magnetic 321,  367 

, ,  Electrostatic 273 

, ,  Practical       368 

Electric  Conductance 334 

Conductors 274,  315,  326 

iu  Parallel 335 

in  Series 335 

Current 304  et  seq. 

,  Energy  of 312,313,369 

,  Heating  Effect  of 329,  330,  339 

,  Induced 371  et  seq. 

,  Magnetic  Effect  of 331,  332,  359  et  seq. 

,  Measurement  of 317,365,369 

,  Steady,  Laws  of 329  et  seq. 

,  Uniformity  of 329 

,  Unit •.    .....     363 

Field,  Attraction  and  Repulsion  in 278 

,  Energy  of 277 

Forces 273,284 

Potential 283 

,  Measurement  of 302 

Resistance 333 

Sparks 284 

Electrical  Machines 277,  294 

Electrification 269 

,  Energy  of 301 

by  Induction 288,  291 

Electro-Chemical  Equivalent 321 

Electrodes 316 

Electrolysis 316 

,  Dissociation  Theory  of 318,  322  et  seq. 

,  Faraday's  Laws  of 316 

Electrolyte 316 

Electro-magnet 361 

Electro-magnetic  Force 361 

Electrometers 302 

Electro- motive  Force  .   • 306 

,  Induced 371 

,  Measurement  of 367 

,  Peltier 309 

,  Source  of 312-315 

— ,  Standard  of ••      315,368 

,  Thomson 310 

Electrophorus •     •     294 

Electro-plating  . 324 

Electroscope 276 


502  INDEX 

Electrostatic  Force,  Law  of 273 

,  Lines  of 279-282 

Induction 288 

Measurements 302 

Element,  Definition  of 5 

Emission  and  Absorption     ...,.- 249,  250 

of  Ether-Waves 247-251,  253 

Energy 75  et  seq. 

,  Conservation  of    /. 76,  200,  262 

,  Intrinsic 200,  226 

,  Kinetic 80,  255 

,  Potential 79 

,  Transfer  of 80,  82,  245  et  seq. 

of  Electric  Currents •>....      312,313 

of  Electrification 301 

of  Electrostatic  Field 277 

of  Heat-Effects 195  et  seq. 

of  Magnetic  Field 369 

Energy-Curve 248 

Engine,  Perfect  Heat- 263 

Equilibrium  of  a  Particle 68 

of  a  Rigid  Extended  Body,  a.  non-parallel  forces 70 

,  6.  parallel  forces 71 

,  c.  couples 72 

,  Stability  of 73 

,  Principle  of  Stable 74 

, ,  in  Dissociation , 243 

, ,  in  Evaporation 237 

, ,  in  Expansion 206,  208,  215 

, ,  in  Fusion 228 

, ,  in  Solution 244 

Equipoteutial  Surfaces 287 

Erg 78,  216 

Ether,  The    .     .    .    .     .     .     ...     .     .....'.     .     .    »'   .      247,267 

,  Effect  of  Matter  upon 268 

Ether-Waves 247  et  seq.,  392  et  seq.,  493 

,  Emission  of       247,  252 

,  Velocity,  Wave-Length,  etc.,  of 248,  394 

— , ,   Measurement  of 398,473,481 

,  Effect  of  Matter  upon 393,  400,  425 

Evaporation,  Latent  Heat  of 236 

,  Laws  of 231  et  seq. 

Expansion 197,  204  et  seq. 

,  Apparent 210 

,  Coefficients  of 204,  206 

,  ,  Measurement  of 210,  211,  215 


INDEX  503 

Expansion  of  Crystals 205 

of  Gases ,    .    .       212-215 

of  Liquids 207-211 

of  Solids 204-207 

,  Hollow 207 

of  Water 208 

Extra  Current  on  Breaking 377 

on  Making 377 

Eye,  The  Human 454 

Eye-Piece 450, 453 

Faraday,  Ice-Pail  Experiment 291 

,  Induced  Currents 371 

,  Laws  of  Electrolysis 316 

Faraday-Tubes 293 

Fixed  Points  of  a  Thermometer 201 

Fizeau,  Measurement  of  Velocity  of  Light 399 

Flames,  Sensitive 157 

Flexure ' 109 

Floating  Bodies ......     119 

Fluids,  Properties  of 4,  129 

Fluorescence 466 

Focal  Lengths 443 

Foci,  Conjugate 166,  416 

Focus,  Real 417,447 

-,  Virtual 443 

Forces 36  et  seq. 

,  Composition  and  Resolution  of 37 

, of  Non-Parallel 61-63 

, of  Parallel 64-66 

,  Resultant 61 

Foucault,  Measurement  of  Velocity  of  Light 399 

Fourier's  Theorem 148 

Fraunhofer  Lines 464,  481 

Free  Path  in  a  Gas 254 

Freezing  Mixtures 230 

Frequency 91,  147 

Fresnel's  Mirrors 472 

Fundamental  Vibration  . 148,  174 

Fusion,  Change  in  Volume 228 

,  Latent  Heat  of 229 

,  Laws  of 226  et  seq. 

Fusion-Point 227 

,  Effect  of  Dissolved  Substances  on 230 

, of  Pressure  on 228 


504  INDEX 

Galvanometer,  Tangent 365 

Constant 366 

Galvanoscope     .     .     . 332 

Gas  at  Rest,  Properties  of 6,  128 

— ,  Elasticity  of 136 

,  Electric  Discharge  through 326 

,  Expansion  of 212 

in  Motion,  Properties  of 134 

,  Internal  Work  in 225 

,  Kinetic  Theory  of  . 255 

,  Mathematical  Law  for  Expansion  of \    .     .  214 

,  Specific  Heat  of 218 

,  "  Standard  Conditions  "of       . 213 

Gas-Constant,  R0 260 

Gay-Lussac's  Law 212 

Geissler-Toepler  Air-Pump 138 

Glow-Lamp 330 

Gram  (Unit  of  Mass) 8 

Gratings,  Plane '..... 479 

Gravitation  due  jto  Earth 96 

,  Universal 97 

Gravity,  Centre  of 99 

, ,  Determination  of 47,  48,  100 

Hall  Effect,  The 386 

Harmonic  Motion :  Rotation 30,  58 

:  Translation ;  ....     27,  59 

Harmony .189 

Heat-Effects 76,  195  et  seq. 

Heat-Energy 195  et  seq. 

,  Flow  of 251 

,  Sources  of 196 

,  Transfer  of 245  et  seq. 

Heating  Effect  of  Electric  Currents .  329,  330,  339 

v.  Helmholtz,  Explanation  of  Harmony 191 

Homogeneous  Waves 397 

Hooke's  Law 104 

Horizontal  Intensity  of  Earth's  Magnetic  Field,  Measurement  of      355,  367 

Horse-Power 82 

Huyghens'  Zones 402,  477 

Hydraulic  Press Ill 

Hygrometric  State 238 

Ice-Calorimeter 222 

Ice-Pail  Experiment,  Faraday's 291 

Images,  Real 417.  448 


INDEX  505 

Images,  Virtual 411,  419 

Impact 34 

Impulse 36 

Incandescent  Electric  Lamp 330 

Incidence,  Angle  of 408 

,  Plane  of 409 

Inclined  Plane 20,  42,  55 

Index  of  Refraction 424 

Induced  Currents 371 

Induction,  Electro-magnetic 371 

,  Electrostatic 288,  291 

,  Magnetic 344 

Induction-Coil 378 

Inertia 2,  31,  32 

,  Centre  of 42  et  seq. 

, ,  Determination  of 47 

, ,  Motion  of 46,  56-58 

,  Relation  between  Weight  and 96 

Intensity  of  Electric  Current 304 

of  Magnetic  Field 348,  355,  357 

of  Sound 161 

Interference  of  Light  Waves 395,  471  et  seq.,  493 

of  Sound  Waves 163 

Interval,  Musical 191 

Intrinsic  Energy 200,  226 

Ions 318 

,  Charges  on 319,  322 

Isothermals 239 

Jets 157 

Joly,  Steam  Calorimeter 223 

Joule,  Internal  Work  in  a  Gas 225 

Kaleidoscope 413 

Kathode 316, 327 

Rays 327 

Kinematics 13  et  seq. 

Kinetic  Theory  of  Evaporation 254 

of  Gases 255  et  seq. 

of  Matter        6,  109,  253 

Kundt,  Measurement  of  Velocity  of  Sound 185 

Lenses,  Converging 445  et  seq. 

,  Diverging 440  et  seq. 

Lever 83 

Leyden-Jar 296 


506  INDEX 

Liebig's  Condenser 234 

Liquefaction  of  Gases 239 

Liquids  at  Rest,  Properties  of 3,  102,  109  et  seq. 

,  Density  of 115,118 

.Expansion  of 207-211 

in  Motion,  Properties  of 121, 122 

Local  Action  in  Primary  Cells 315 

Loops 170 

Machines .       83 

Magnetic  Effect  of  Electric  Currents 331,  341,  359  et  seq. 

Elements  of  Earth 357 

Eield 340 

,  Attraction  and  Repulsion  in 341,  346,  350 

,  Energy  of 350 

,  Intensity  of 348,  355 

,  Measurement  of .      352-355 

Force,  Law  of 345 

,  Lines  of 347 

Induction % 348-350 

Meridian  of  Earth 356 

Moment 351 

,  Measurement  of 352-355 

Needle 331,340 

,  Astatic .     .     331 

,  Vibrations  of .     ...     .     352 

Permeability 346,  348 

Polarity 331, 342 

Pole,  Unit 345 

Potential 347 

Substances 341 

Magnetism,  Ampere's  Theory  of 362 

,  Effect  of  Heat  on 343 

,  Molecular  Nature  of 343-345 

Magnets,  Properties  of 341  et  seq. 

Magnifying  Power 418.  444,  450 

Major  Triad 188 

Manometer,  Closed 136 

,  Open 133 

Mass 7 

,  Measurement  of 31,32,85,97 

Matter  (Properties  of) 2  et  seq. 

,  Conservation  of 7 

,  Divisibility  of 4 

,  Forms  of 3,  102 

,  Fourth  State  of      ....  254,  327 


INDEX  507 

Matter,  Kinetic  Theory  of , 6,  253 

,  Quantity  of 6 

Mechanical  Advantage 83 

Mechanical  Equivalent  of  Heat 262 

Mechanics 9 

Membranes,  Vibrations  of 181 

Metacentre .' 120 

Metre 7 

Microphone 384 

Microscope,  Compound 450 

,  Simple 448 

Mirror,  Concave 415  et  seq. 

,  Convex 1 419 

,  Plane 407,  410 

,  Rotating 409 

Mirrors,  Inclined 412 

Molecular  Changes,  as  Heat-Effect 199,  225  etseq. 

Weight 259 

Molecules 5 

Moment  of  a  Force 51 

of  Inertia       51 

of  Momentum 50 

,  Conservation  of 50 

Momentum,  Angular 51  et  seq. 

, ,  Conservation  of 50,  53 

,  Linear 33,  257 

, ,  Conservation  of 33-35 

Moon,  Motion  of 98 

Motion 7, 13 

,  Newton's  Laws  of 38 

Motors,  Electric 363, 383 

Musical  Notes 158,188 

Scales 190 

Newton's  Laws  of  Motion 38 

Rings 477 

Nicol's  Prism 485 

Nodes 169 

Noises 147 

Nuclei,  Effect  of 227,234 

Objective,  —  Object-Glass 450,  453 

Octave 188 

Ohm,  Unit  of  Electric  Resistance 368 

Ohm's  Law,  Sound  Sensation 159 

,  Steady  Electric  Currents 333 


508  INDEX 

Organ-Pipes,  Closed 178 

,  Open 179 

Osmosis 122 

Osmotic  Pressure 122 

Page  Effect 345 

Parallelogram  of  Forces 4j 

Partial  Vibrations 148,  174,  189 

Particles,  Reflection  by  Fine 466 

Peltier  E.M.F .  309 

,  Hydrostatic  Analogy 308 

Pencil  of  Waves 412 

Pendulum,  Compound m  100 

,  Simple 58 

Penumbra 405 

Period    .     . .     .     .     .  26,  28,  147 

,  Measurement  of 150 

Phosphorescence 466 

Photometer 394 

Pin-Hole  Images 405 

Pitch  of  Sounds 159 

,  Limits  of 157 

Plane  of  Incidence 409 

of  Polarization 489 

Plasticity ..-..'.. 103 

Plate,  Flat,  Refraction  through    .     .  ' 428,  432 

Plates,  Vibrations  of 181 

Points,  Effect  of,  on  Electric  Discharge 290 

Polarization,  Angle  of 489 

,  Colors  due  to 493 

,  Electric 275,  314 

,  Plane  of 489 

by  Absorption 485,  490 

by  Double  Refraction 489 

by  Reflection .     .     .    ' 487 

Polarized  Waves,  Circularly 492 

,  Elliptically 492 

,  Plane 488 

,  Interference  of 493 

Porosity , 4 

Potential,  Electric 283 

Power 82 

Pressure 110 

,  Atmospheric 129 

,  Centre  of .  115 

in  a  Bubble  or  Drop 125 


INDEX  509 

Pressure  in  a  Gas . 129,  135 

on  Kinetic  Theory 257 

,  Measurement  of 132 

in  a  Liquid 109-115 

Primary  Electric  Cell 310 

,  Hydraulic  Analogy 314 

Prism  :  Angle,  Edge,  Face 429 

Pulley 85 

,  Differential 87 

Pump,  Air 137-140 

,  Water 131 

Quality  of  Sound 161 

Quarter-wave-plate 493 

Radiation       247  et  seq. 

and  Absorption       250 

Radiometer,  Crookes' 254 

Ratio  of  Specific  Heats  of  a  Gas -     137,  154,  183,  219 

Ray  of  Light 403 

Rectilinear  Propagation  of  Light 401 

Reflection,  Angle  of 409 

from  a  Fixed  End       169 

from  a  Free  End 171 

of  Ether- Waves 250,  407  et  seq. 

of  Light  :  Concave  Surface 415  et  seq. 

:  Convex  Surface .419 

:  Fine  Particles 466 

:  Plain  Surface '407 

of  Sound 167  et  seq. 

,  Total I     .     .     427 

Refraction  of  Light :  Concave  Surface 435,  437,  438 

:  Convex  Surface 435,  438,  439 

:  Plane  Surface 423,  431 

:  Plate  with  Parallel  Faces 428,  432 

:  Prism       429, 433 

,  Angle  of 424 

,  Double , 483  et  seq. 

,  Index  of 424 

, ,  Measurement  of 427,  430,  432 

of  Sound 166 

Resistance,  Electric 333 

, ,  Measurement  of 337 

Resolving  Power 452,  481 

Resonator 162 

Resultant :  Non-parallel  Forces 61 


510  INDEX 

Resultant :  Parallel  Forces 64 

Right-Handed-  Screw  Law 33 1?  495 

Rigidity 102,  105 

,  Coefficient  of 106 

Rods,  Vibrations  of 177 

Rontgen  Rays 328 

Rotation,  Motion  of 14,21,28 

around  a  Fixed  Axis 28 

of  the  Plane  of  Polarization  by  Magnetic  Field      ....      386,  495 

by  Quartz 495 

Scales,  Musical 190 

Screw 88 

Second,  Mean  Solar 9 

Semi-tone 191 

Sextant 410 

Shadows 403 

Sharps  and  Flats 191 

Shielding  from  Electrostatic  Action 294 

from  Magnetic  Action 350 

Sine-Curve 148 

Siphon 131 

Siren 150 

Solenoid 360 

Solid  (Properties  of) 3,  102,  104 

,  Density  of 118 

,  Expansion  of 204 

Solution. 243 

Sounds 145,157 

,  Intensity 161 

,  Pitch  • 159 

,  Quality 161 

Sound  Vibrations 145 

,  Frequency 147 

,  Intensity 147 

Waves 151 

,  Form 155 

,  Intensity 153 

,  Velocity 153,  183  et  seq. 

.Wave-Length .'    .     .   152,169,171 

,'  Wave-Number 153 

Sounding-Body 145,  173  et  seq. 

Specific  Heat 217 

of  Gases 218 

,  Measurement  of :  Mixtures 220 

, :  Condensation  of  Steam  .  ,    223 


INDEX  511 

Specific  Heat,  Measurement  of :  Melting  Ice 222 

, :  Flow 223 

Spectroscope,  Direct  Vision 460 

,  Prism 461 

Spectrum 457  et  seq. 

,  Absorption 463 

,  Continuous 249,  462 

,  Line 463 

,  Pure  and  Impure 456 

,  Solar 464 

Speed,  Angular 22 

,  Linear 16 

Spheroidal  State 238 

Spiral  Spring 31,  37,  107 

Sprengel  Air-Pump 138 

Stationary  Vibrations 170,  171 

Statistics,  Principle  of 255 

Steam  Calorimeter 223 

Straight  Line,  Motion  in  a 23 

Strain 104 

,  Electric 269,  272 

Strength  of  a  Couple      . 67 

Stress 104 

Stride  in  Vacuum-Tubes 327 

Sublimation 242 

Summational  Notes 165 

Surface  Absorption 465 

Surface-Color ' 469 

Surface-Tension 124  et  seq. 

Sympathetic  Vibrations 163 

Telephone      .     .     .* 383 

Telescope 452 

,  Power  of        452 

Temperature 200 

,  Change  in  ;  as  Heat-Effect 198,  216  et  seq. 

of  Inversion 310 

on  Kinetic  Theory 255 

Tempered  Scale 191 

Thermal  Unit 216 

Thermo-electric  Couple .    310 

Current 307-310 

Thermometer 201 

Scale 202 

Thermodynamics,  First  Principle 262 

,  Second  Principle 262,  263 


512  INDEX 

Thermopile 310 

Thomson  E.  M.  F 310 

Tides 99 

Tone,  Musical 159,  191 

Top,  Motion  of  a 29,  56 

Torsion 106 

Tourmaline 485 

Transformer 378 

Translation,  Motion  of 13,  14,  23,  33 

Transmission  of  Ether- Waves 251 

Tuning-Fork 177 

Ultra-Red  Light 457 

Ultra- Violet  Light 457 

Umbra 405 

Units,  Electro-magnetic 367 

,  Electrostatic 273 

,  Mechanical,  C.  G.  S 7-9,  101 

,  Practical  Electrical .  368 

"  •"  / 

Valency 316 

Vapors 231  et  seq. 

,  Saturated 231 

, ,  Equilibrium  of 231 

, , ,  over  Curved  Surfaces 232 

, ,  Laws  of 231,  240 

, ,  Pressure  of :  Dynamical  Method .     .     234 

, , :  Statical  Method 233 

,  Unsaturated 231 

Velocity,  Angular 21 

,  Composition  and  Resolution 17,  21,  22 

,  Linear       15 

of  Ether- Waves 398 

of  Sound- Waves 183  et  seq. 

Vertical  Circle,  Motion  in  a 81 

Vibration,  Harmonic 27,  147 

,  Stationary 170,171 

,  Sympathetic 163 

of  Bells 181 

of  Columns  of  Gas :  Closed 178 

:  Open 179 

of  Plates  and  Membranes 181 

of  Rods,  Longitudinal 177 

,  Transverse 177 

of  Stretched  Cord,  Longitudinal , 176 

,  Transverse  1 73-1 75 


INDEX  513 

Virtual  Images 411,419 

Viscosity         103 

Voice,  The  Human 182 

Volt  (Unit  of  E.  M.  F.) 368 

Voltaic  Cell 311-314 

Voltameter 317 

Volume,  Change  in  ;  as  Heat-Effect      ........     197,2Q±  et  seq. 

,  Work  required  to  Produce 120,134,198,218 

Water,  Expansion  of 208 

Water-Equivalent 221 

Watt  (Unit  of  Power) 82 

Wave-Normal , 403 

Waves,  Composition  of 154 

,  Intensity  of 93 

,  Longitudinal  92,151 

,  Primary  .... 95 

,  Secondary 95,  402 

,  Transverse 92,169,488,493 

,  Velocity  of 94,  153 

Wave-Front 94 

Wave-Length 91 

of  Light,  Measurement  of 473,  481 

Wave-Motion 90-95 

Wave-Number 91 

Weber,  Theory  of  Diamagnetism 385 

Weight 2,  96 

,  Relation  between  Inertia  and 96 

Wheatstone's  Bridge .  337,  338 

Whirling  Table  40 

Whispering-Gallery 166 

Work 75  et  seq. 

— ,  Measure  of 76-80 

—  necessary  to  change  Volume  of  a  Fluid      ....     120,134,198,218 

Young's  Interference  Experiment 395 

Modulus  107,  176 


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